Calculate the time required to initiate the transformation

Question

The values of the parameters n and K of Johnson-Mehl-Avrami equation for a phase transformation at 4000
 and 3600 C are given as follows.

 

Temperature                                   n                                                  K(per sec^2)

400                                                2.0                                                   6.4x10^-3

360                                                2.0                                                   0.9x10^-3

 

Calculate the time required to initiate the transformation (i.e., time required for X = 0.01) and the time
required to complete the transformation (i.e., X = 0.99) at 4000 and 3600C.

 

Answer 1

To start, we will state the equation given in the problem statement.

Johnson-Mehl-Avrami Equation: x(t) = 1 - exp[-K*tⁿ]

1. K: reaction rate constant
2. t: time
3. n: Avrami exponent
4. x(t): fraction of transformed material as a function of time

Time Required To Initiate Transformation (X = 0.01)

As with many advanced science problems, this really just turns into an algebra problem once you set it up. To calculate the time required at X = 0.01, we simply need to substitute X = 0.01 and solve for t. For simplicity, we will first solve the equation for t, then substitute X = 0.01 to find the time required.

Step 1: Isolate the exponential term. Subtract x from both sides, and add the exponential term to both sides.

exp[-K*tⁿ] = 1 - x

Step 2: We will take the natural log of both sides of the equation, as ln(e^x) = x.

ln(exp[-K*tⁿ]) = ln(1 - x)

-K*tⁿ = ln(1 - x)

Step 3: Now we need to isolate t, as this is what we are solving for. We will first divide both sides by -K, then raise both sides of the equation to the power of 1/n in order to get t by itself.

tⁿ = (-1/K)*ln(1 - x)

(tⁿ)^1/n = [(-1/K)*ln(1 - x)]^1/n

t(x) = [(-1/K)*ln(1 - x)]^1/n

Now, we have a transformed the function from finding x as a function of t to finding t as a function of x, making the rest of the problem trivial.

At T = 4000 °C: t(0.01) = [(-1/(6.4*10^-3)*ln(1 - 0.01)]^1/2 = 1.253142042 ≈ 1.3 seconds (the units of K are s^-2, so the time unit is seconds)

At T = 3600 °C: t(0.01) = [(-1/(0.9*10^-3)*ln(1 - 0.01)]^1/2 = 3.341712112 ≈ 3.3 seconds

Time Required to Complete the Transformation (x = 0.99)

At T = 4000 °C: t(0.99) = [(-1/(6.4*10^-3)*ln(1 - 0.99)]^1/2 = 26.82457533 ≈ 26.8 seconds

At T = 3600 °C: t(0.99) = [(-1/(0.9*10^-3)*ln(1 - 0.99)]^1/2 = 71.53220088 ≈ 71.5 seconds

Final Answers:

For T = 3600°C:

1. t(0.01) = 3.3 s
2. t(0.99) = 71.5 s

For T = 4000°C:

1. t(0.01) = 1.3 s
2. t(0.99) = 26.8 s