MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Kinetic Processes in Materials

3.21 Spring 2001

S. M. Allen and W. C. Carter

Department of Materials Science and Engineering

Massachusetts Institute of Technology

77 Massachusetts Ave.

Cambridge, MA 02139

Problem Set 4: Due Fri. March 16, Before 5PM in 13-5049

Exercise 4.1
Use the interdiffusivity data provided in Exercise 1.4 (The last problem of Problem Set 2) to compute the composition vs. distance profile in a Au-Ni diffusion couple held at 900$^\circ$ C for $10^6$ s. Assume that the diffusion couple is infinite with initially pure Au for $x<0$ and Ni for $x>0$.

Exercise 4.2
A sealed spherical metal tank with 5 mm wall thickness and 20 cm diameter holds hydrogen gas maintained at an internal pressure of $5 \times 10^{6}$ MPa.

The equilibrium solubility of hydrogen dissolved in the metal is $2.5 \times 10^6$ cm$^3$(STP) g$^{-1}$ (note this is at $10^5$ MPa). Assume Sievert's law holds.

If the diffusivity of hydrogen in the metal is $5 \times 10^{-10}$ m$^2$ s$^{-1}$, calculate the steady-state rate of hydrogen loss (cm$^3$(STP) s$^{-1}$) from the pressurized tank when the exterior of the tank is at $10^5$ MPa of air.

Exercise 4.3
A 0.25 inch thick plate of 1080 plain-carbon steel is tightly clamped to a pure iron sheet 0.125 inch thick, then hot-isostatically pressed (HIPed) at 1000$^\circ$ C for 10 hrs. Assuming a carbon diffusivity of $3\times 10^{-11}$ m$^2$ s$^{-1}$, and that there is no carbon loss from either surface of the material to the atmosphere, calculate the expected carbon content at both surfaces of the pressed assembly.

Exercise 4.4
Solve the isotropic, uniform diffusivity diffusion equation on an infinite cylindrical wedge with a vertex angle of $2 \pi/5$: $0 < \theta < 2 \pi/5$, $-\infty < z < \infty$, $0 < r < \infty$. with the following initial

$$c(r,\theta, x, t) = \left\{ \begin{array}{ll} C_0 & 0 < r < ... ..._i < r < R_o\\ C_0 & R_o < r < \infty\\ \end{array}\right.$$

and boundary conditons

$$J(r,\theta = 0, z, t) = J(r,\theta = 2 \pi/5, z, t) = J(r = \infty,\theta,z, t) = J(r,\theta,z= \pm \infty, t) = 0$$