1. Consider a steel tank containing hydrogen at 10 atm pressure with a vacuum on the outside.
Take the solubility of hydrogen in steel at the inner surface in equilibrium with hydrogen at 10
atm to be 10-2 g/cm2, and the diffusion coefficient for hydrogen to be 10-5 cm2/sec
a. Calculate the flux of hydrogen through a wall of thickness 1 mm, in grams per square
centimeter per second
b. The diffusion coefficient of carbon in austenite can be approximated by the equation
= 0.2 exp
i.
ii.
iii.
−138,600 /
⁄
Evaluate Dc for 920 oC
How long does it take for the composition C0.5 to penetrate 1 mm at this
temperature? How long for 2 mm?
What temperature of anneal is required to double the penetration in a given time?
2. An Fe-C alloy is placed in a decarburizing atmosphere which holds the surface concentration
at zero carbon content
a. Draw C(x) after some finite time
b. Directly beneath this, suing the same x axis plot J(x)
⁄ ! versus x, again using the same x axis. Note that
c. Directly beneath this, sketch
the surface concentration is kept fixed
3. A report put out by a junior engineer on casting homogenization asserts that he can treat the
homogenization of the casting with the equation
(#,$) = &' exp(!) exp(−( )
)
a. Show that this is or not a solution to the diffusion equation
b. The engineer asserts that the time required to get homogenization is ( = 1⁄√
Discuss the plausibility of this
4. Someone opens a bottle of pungent perfume about 10 m away from you and you notice the
smell in a few seconds. Could the scent have diffused through the air to your nose or was it
transported by convection? (D in air is 1 cm2/sec)
5. Given that
+ =
16 Γ+ , consider the diffusion of vacancies
a. Show that , = .' /√2 for a vacancy jumping in an fcc lattice
b. Let ΔSm/R = 2, and v = 1013 sec. Calculate Do for vacancies. (Take ao = 4Å)
c. If ΔHm = 95 KJ/mole, calculate Dv for vacancies at 800 oC
6. A piece of 0.1 % C steel is to be carburized at 930 oC until the carbon content is raised to 0.45%
C at a depth of 0.05 cm. The carburizing gas holds the surface at 1 % carbon for all t > O. If D
= 1.4 x 10- 7 cm2/s for all compositions
(a) Calculate the time required at the carburizing temperature.
(b) What time is required at the same temperature to double this amount of penetration? (c) If
D = 0.27 exp(-17,400/T) cm2/s, what temperature increase is required to get 0.45% C at a depth
of 0.1 cm in the same time as 0.05 cm was attained at 9300 C?
7. A thick-walled steel pressure vessel in an oil refinery contains high pressure hydrogen for a
long time. To avoid hydrogen cracking on cooling the vessel is to be held at temperature withno
hydrogen inside it until most of the H has diffused out. As boundary conditions take for
c(x,t):c(x,0) = co x/h at t = 0
c(0,t) = 0 = c(h,t) at t > O.
(a) Derive c(x,t) for t > O. (b) If D = 5.8 X 10- 4 cm2/s, and h = 25 cm, how long does
it take to get the average concentration, ̅ C, reduced to 0.1 co?
8. In hydrogen gas at 1 atm and 25 oC, the average molecular velocity is 1.3 x 105 cm/s, and the
mean free path is 1.9 x 10-5 cm. Calculate the diffusion coefficient of the gas. (Take the average
velocity to be the same as the root-mean-square velocity.)
9. The force on an ion in a solution is the charge q times the negative of the potential gradient
∂Φ⁄∂x
a. Write the equation for the mean velocity of the ion, if the mobility is D/kT.
b. Assume that the concentration gradient acts independently of the electrical force so that
the net force on the ion is the sum of the two forces. Show that the ion flux is then
=
1
(
2 ln
2Φ
+
)
2!
2!
c. If the current is vq, show that the electrical resistivity of the solution is kT/Dcq2 (Take
dc/dx = 0)
10. If at Z = 0 a quantity of solute is located at the point r = 0 in a three dimensional medium, the
concentration of solute at any point r from the origin, after time t, is given as
(6, ( ) =
a.
7
8
$ -9
!: ;
<= 9
>?$
@
Give the probability (normalized to 1) of finding an atom in a spherical shell dr thick at
a distance r from the origin
b.
What is the mean square value of r that is 6AAA for the solute after time t. Using the result
of part (b) and random walk equation
Show that = 1-6 Γ a where Γ = D-(
Due for submission on final examination date