GATE 2026 MT – Question 60
During carburizing of steel, the surface concentration is maintained at 1.4 wt.% carbon. The carbon diffusivity at $950^\circ\mathrm C$ is $6.25\times10^{-11}\ \mathrm{m^2/s}$. At this temperature, the time required to increase the carbon concentration from an initial 0.2 wt.% to 0.8859 wt.% at a depth of 0.2 mm is ________ seconds. Approximate to the nearest integer and use the nearest error-function value from the table.
$z=0.3$: $\operatorname{erf}(z)=0.3268$.
$z=0.4$: $\operatorname{erf}(z)=0.4284$.
$z=0.5$: $\operatorname{erf}(z)=0.5205$.
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Correct answer: 990 to 1010
Explanation
For constant surface concentration in a semi-infinite solid, $\operatorname{erf}(z)=(C_s-C_x)/(C_s-C_0)=(1.4-0.8859)/(1.4-0.2)=0.4284167$. The nearest table value corresponds to $z=0.4$.
Since $z=x/(2\sqrt{Dt})$, $t=x^2/(4Dz^2)=(0.0002)^2/[4(6.25\times10^{-11})(0.4)^2]=$ **1000 s**.