Iron powder is compacted at room temperature to 75% of its theoretical density. The as-pressed iron compact is sintered in inert-gas atmosphere at \(1200^{\circ}\text{C}\) to 90% of its theoretical density. Assuming isotropic shrinkage, the linear shrinkage (in percent) undergone by the iron compact during sintering (rounded off to one decimal place) is ______ %.
Show Hint
Use \(V\propto 1/D\) for constant mass, then \(L_1/L_0=(D_0/D_1)^{1/3}\) since the shrinkage is isotropic.
Step 1: Work with the green-to-sintered volume ratio instead of sintered-to-green.
Since the compact carries the same mass throughout, $V\propto 1/D$ where $D$ is the fractional density. Take the ratio the other way round:
\[ \frac{V_{green}}{V_{sintered}}=\frac{D_{sintered}}{D_{green}}=\frac{0.90}{0.75}=1.2 \]
Step 2: Take the cube root to get the length ratio.
For isotropic shrinkage, length scales as the cube root of volume, so
\[ \frac{L_{green}}{L_{sintered}}=(1.2)^{1/3} \]
Working this out, $(1.2)^{1/3}\approx1.0627$.
Step 3: Flip this to get the sintered fraction.
\[ \frac{L_{sintered}}{L_{green}}=\frac{1}{1.0627}\approx0.9410 \]
Step 4: Write the shrinkage as a fraction lost.
\[ \text{Linear shrinkage}=\left(1-\frac{L_{sintered}}{L_{green}}\right)\times100=(1-0.9410)\times100 \]
Step 5: Finish the calculation.
\[ \text{Linear shrinkage}\approx5.9\% \]
This matches the direct calculation, confirming the compact shrinks by about $5.9\%$ in each linear direction while densifying from $75\%$ to $90\%$ of theoretical.
Step 6: Why the two routes agree.
Both paths use the same physics, that mass is conserved while volume falls as density rises, and that isotropic shrinkage lets us move between volume and length with a cube or a cube root. Starting the ratio as green-over-sintered instead of sintered-over-green just flips which fraction we compute first, the final linear shrinkage number comes out the same either way.
\[ \boxed{5.9\%} \]
