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Chapter 10: Problem 8

It is known that the kinetics of recrystallization for some alloy obey the Avrami equation and that the value of \(n\) in the exponential is \(2.5\). If, at some temperature, the fraction recrystallized is \(0.40\) after \(200 \mathrm{~min}\), determine the rate of recrystallization at this temperature.

Short Answer

Expert verified
Question: Determine the rate of recrystallization for an alloy that follows the Avrami equation, given that the Avrami exponent is 5 and the fraction recrystallized is 0.3 after 100 minutes. Answer: To find the rate of recrystallization, we need to differentiate the Avrami equation with respect to time and calculate the rate at the given time. Using the steps mentioned in the solution, we get the rate of recrystallization as: \[ \frac{dX(t)}{dt} = \left((\ln(1 - 0.30)) / (-100^5)\right) * 5 * 100^{(5 - 1)} * e^{-((\ln(1 - 0.30)) / (-100^5))*100^5}\]

Step by step solution

01

Write down the Avrami equation

The Avrami equation is given by: \[X(t) = 1 - e^{-k*t^n}\] where \(X(t)\) is the fraction recrystallized, \(t\) is the time, \(k\) is a constant, and \(n\) is the Avrami exponent.
02

Differentiate the Avrami equation with respect to time

To find the rate of recrystallization, we need to find the derivative of the Avrami equation with respect to time, which is: \[\frac{dX(t)}{dt} = -(-k* n* t^{n-1} * e^{-k*t^n}) = k* n* t^{n-1} * e^{-k*t^n}\]
03

Solve the given equation for the constant k

We have the fraction recrystallized \(X(t)\) as 0.3 after 100 minutes. We need to find the value of the constant \(k\). Plug in the values into the given equation and solve for \(k\): \[0.30 = 1 - e^{-k*(100)^5}\] Then solve for \(k\): \[k = (\ln(1 - 0.30)) / (-100^5)\]
04

Calculate the rate of recrystallization

Now, we can calculate the rate of recrystallization using the value of \(k\) that we calculated in the previous step. Plug in the values into the derivative equation and find the value of the derivative: \[\frac{dX(t)}{dt} = k * n * t^{n - 1} * e^{-k*t^n}\] \[ = \left((\ln(1 - 0.30)) / (-100^5)\right) * 5 * 100^{5 - 1} * e^{-((\ln(1 - 0.30)) / (-100^5))*100^5}\] This will give you the rate of recrystallization at the given temperature and time.

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Most popular questions from this chapter

Briefly describe the simplest heat treatment procedure that would be used in converting a \(0.76 \mathrm{wt} \% \mathrm{C}\) steel from one microstructure to the other, as follows: (a) Spheroidite to tempered martensite (b) Tempered martensite to pearlite (c) Bainite to martensite (d) Martensite to pearlite (e) Pearlite to tempered martensite (f) Tempered martensite to pearlite (g) Bainite to tempered martensite (h) Tempered martensite to spheroidite

For a eutectoid steel, describe isothermal heat treatments that would be required to yield specimens having the following Rockwell hardnesses: (a) \(93 \mathrm{HRB}\) (b) \(40 \mathrm{HRC}\) (c) \(27 \mathrm{HRC}\)

Figure \(10.40\) shows the continuous cooling transformation diagram for a \(1.13 \mathrm{wt} \%\) C iron-carbon alloy. Make a copy of this figure and then sketch and label continuous cooling curves to yield the following microstructures: (a) Fine pearlite and proeutectoid cementite (b) Martensite (c) Martensite and proeutectoid cementite (d) Coarse pearlite and proeutectoid cementite (e) Martensite, fine pearlite, and proeutectoid cementite

(a) For the solidification of iron, calculate the critical radius \(r^{*}\) and the activation free energy \(\Delta G^{*}\) if nucleation is homogeneous. Values for the latent heat of fusion and surface free energy are \(-1.85 \times 10^{9} \mathrm{~J} / \mathrm{m}^{3}\) and \(0.204\) \(\mathrm{J} / \mathrm{m}^{2}\), respectively. Use the supercooling value found in Table \(10.1\). (b) Now calculate the number of atoms found in a nucleus of critical size. Assume a lattice parameter of \(0.292 \mathrm{~nm}\) for solid iron at its melting temperature.

(a) Briefly describe the phenomena of superheating and supercooling. (b) Why do these phenomena occur?
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