Fick’s laws of diffusion (1855)

Adolf Fick, a 26-year-old prosector in anatomy at Zürich, took Fourier’s law of heat conduction and applied it to salt dissolving in water: the flux is proportional to the concentration gradient, \(J = -D\,\partial C/\partial x\) (fick_flux()), and conservation gives the second law \(\partial C/\partial t = D\,\partial^2 C/\partial x^2\). Into a thick steel part whose surface is held at a carbon content \(C_s\), the second law gives the error-function profile (semi_infinite_concentration()) that times every carburizing furnace. The example computes the case depth of a gear and the furnace time it needs.

import matplotlib.pyplot as plt
import numpy as np

from engineeringkit.materials import diffusion_coefficient, diffusion_time, fick_flux, semi_infinite_concentration

Steady flux through a plate

Fick’s first law: carbon at 1.2 and 0.8 kg/m^3 at depths 5 and 10 mm in an iron plate with \(D = 3\times10^{-11}\) m^2/s.

J = fick_flux(3e-11, (0.8 - 1.2) / (10e-3 - 5e-3))
print(f"flux {J:.2e} kg/(m^2 s)")
assert np.isclose(J, 2.4e-9)

flux 2.40e-09 kg/(m^2 s)

Carburizing a gear

Carbon diffuses in austenite with \(D_0 = 2.3\times10^{-5}\) m^2/s and \(Q_d = 148\) kJ/mol. A 0.25 wt% C steel in a 1.2 wt% atmosphere must reach 0.80 wt% at 0.5 mm. Doubling the depth takes four times as long, because time enters only through \(x/\sqrt{Dt}\).

C0, Cs, target, depth = 0.25, 1.20, 0.80, 0.5e-3
for T in (900.0, 950.0, 1000.0):
    D = diffusion_coefficient(2.3e-5, 148e3, T + 273.15)
    t = diffusion_time(depth, target, D, C0, Cs)
    print(f"{T:.0f} C: D = {D:.2e} m^2/s, {t / 3600:.1f} h")
D = diffusion_coefficient(2.3e-5, 148e3, 950 + 273.15)
t = diffusion_time(depth, target, D, C0, Cs)
assert np.isclose(diffusion_time(2 * depth, target, D, C0, Cs), 4 * t)

x = np.linspace(0, 2e-3, 200)
fig, ax = plt.subplots(figsize=(6, 3.8))
for hours in (1, 4, 16):
    ax.plot(1e3 * x, semi_infinite_concentration(x, hours * 3600, D, C0, Cs), label=f"{hours} h")
ax.plot(1e3 * depth, target, "ko", label=f"0.80% at 0.5 mm, {t / 3600:.1f} h")
ax.set(xlabel="depth [mm]", ylabel="carbon [wt%]", title="Carburizing at 950 C")
ax.legend()
fig.tight_layout()
assert np.isclose(semi_infinite_concentration(depth, t, D, C0, Cs), target)

plt.show()

900 C: D = 5.92e-12 m^2/s, 19.1 h
950 C: D = 1.10e-11 m^2/s, 10.2 h
1000 C: D = 1.95e-11 m^2/s, 5.8 h