The homogenized Al-Mg alloy is suddenly quenched and held isothermally:
A) From 400C to 250C
B) From 400C to 200C
How long would it take to regrow an isolated particle of Al3Mg2 to a diameter of 15 microns, assuming that B(Beta) particles nucleate (t=0) from the matrix with an initial diameter of 10^-3 mikrometer, and that the diffusitivity at 250C and 200C is D=1x10^-11 cm^2/sec?
Please consider both cases of quenching at 250C and 200C, respectively.
Don't forget to comment on your results.
To solve this problem, we need to calculate the time it takes to regrow the isolated particle of Al3Mg2 to a diameter of 15 microns under two different quenching conditions: from 400C to 250C and from 400C to 200C. We will use the diffusitivity values at 250C and 200C given in the question, which are D = 1x10^-11 cm^2/sec for both cases.
Since the problem mentions that the B (Beta) particles nucleate from the matrix at t=0, we can assume that the regrowth of the particle occurs by the diffusion of atoms in the matrix towards the particle surface.
The rate of growth of the particle can be described by the equation:
r = (3D * t)^0.5
where r is the radius of the particle, D is the diffusitivity, and t is time.
To find the time it takes to reach a diameter of 15 microns (or a radius of 7.5 microns), we can rearrange the equation as:
t = (r^2) / (3D)
Let's calculate the time for both cases:
A) Quenching from 400C to 250C:
Given:
Initial radius, r = 10^-3 mikrometer
Final radius, r_final = 15/2 micrometer = 7.5 micrometers
Diffusitivity, D = 1x10^-11 cm^2/sec
Converting the units:
Initial radius, r = 10^-8 cm
Final radius, r_final = 7.5x10^-4 cm
Using the equation, we can calculate the time:
t = (r_final^2) / (3D)
t = (7.5x10^-4)^2 / (3 * 1x10^-11)
t = 2.5x10^-7 / (3 * 1x10^-11)
t ≈ 2.78x10^4 seconds
B) Quenching from 400C to 200C:
Given:
Initial radius, r = 10^-3 mikrometer
Final radius, r_final = 15/2 micrometer = 7.5 micrometers
Diffusitivity, D = 1x10^-11 cm^2/sec
Converting the units:
Initial radius, r = 10^-8 cm
Final radius, r_final = 7.5x10^-4 cm
Using the equation, we can calculate the time:
t = (r_final^2) / (3D)
t = (7.5x10^-4)^2 / (3 * 1x10^-11)
t = 2.5x10^-7 / (3 * 1x10^-11)
t ≈ 2.78x10^4 seconds
Commentary:
The calculated times for both cases of quenching (at 250C and 200C) are the same: approximately 2.78x10^4 seconds. This suggests that the temperature difference between 250C and 200C does not have a significant impact on the regrowth time of the particle.
It is important to note that the calculations assume ideal conditions and do not consider any other factors that may affect the diffusion or growth of the particles.
Furthermore, it is always recommended to validate these calculations with experimental data or consult relevant research articles for more precise results in real-world scenarios.