Question

Q5. A hypothetical metal alloy has a grain diameter of 2.4 x 10-2 mm. After a heat treatment at 575°C for 500 min, the grain diameter has increased to 7.3 x 10-2 mm. Compute the time required for a specimen of this same material (i.e., d0 = 2.4 x 10-2 mm) to achieve a grain diameter of 5.5 x 10-2 mm while being heated at 575°C. Assume the n grain diameter exponent has a value of 2.2. (10 points)

Answer 1

Time is 244.89 minutes

Explanation:

Given:

Hypothetical diameter, d₀ = 2.4 X 10⁻² mm

Increase in diameter, d = 7.3 X 10⁻² mm

d₀ = 2.4 X 10⁻² mm

Time, t = 500 min

To solve K:

$K = \frac{d^n - d_o^n}{t}$

On substituting the value:

$K = \frac{(7.3 X 10^-^2)^2^.^2 - (2.4 X 10^-^2)^2^.^2}{500} \\\\\\K = 5.8 X 10^-^6 mm^2^.^2/min$

From the value of K, t can be calculated as:

$t = \frac{d^2^.^2 - d_o^2^.^2}{K} \\\\t = \frac{(5.5 X 10^-^2)^2^.^2 - (2.4 X 10^-^2)^2^.^2}{5.8 X 10^-^6} \\\\t = 244.89 min$