A hypothetical metal alloy has a grain diameter of 2.4 × 10−2 mm. After a heat treatment at 575°C for 500 min, the grain diameter has increased to 5.6 × 10−2 mm. Compute the time required for a specimen of this same material (i. e., d0 = 2.4 × 10−2 mm) to achieve a grain diameter of 5.5 × 10−2 mm while being heated at 575°C. Assume the n grain diameter exponent has a value of 2.2.

The time to achieve a grain diameter of 5.5 × 10⁻² mm is 477 minutes

Explanation:

We need to first solve for K

Given that:

grain diameter exponent (n) = 2.2

grain diameter (d) = 5.6 × 10⁻² mm = 5.6 × 10⁻⁵ m

d₀ =  2.4 × 10⁻² mm =  5.6 × 10⁻⁵ m

t = 500 min

= 7.48 × 10⁻¹³ m/min

From K we can determine the time required based on the desired diameter

grain diameter (d) = 5.5 × 10⁻² mm = 5.5 × 10⁻⁵ m

= 477 min

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