Carburising Case Depth Calculator — Fick’s Second Law Diffusion Model

The case depth calculator on this page uses the Fick’s Second Law semi-infinite slab solution to predict the carbon or nitrogen concentration profile and the effective case depth for gas carburising, vacuum carburising, gas nitriding, and ferritic nitrocarburising (FNC). Enter your process parameters to obtain a full concentration-vs-depth profile, effective case depth (ECD), total case depth (TCD), and a step-by-step worked solution showing every intermediate calculation.

Fick’s Second Law: Derivation and Solution

Carbon and nitrogen diffusion in steel is a thermally activated solid-state process governed by Fick’s laws. In a carburising or nitriding operation the relevant geometry is a semi-infinite slab: the steel component is treated as infinite in depth compared with the case depth of interest, and the surface condition is maintained at a constant concentration C_s throughout the process.

Fick’s Second Law

Fick's Second Law (1D, constant D):

  ∂C/∂t = D × ∂²C/∂x²

Where:
  C = concentration at depth x and time t  [wt%]
  D = diffusivity  [m²/s]
  x = depth below surface  [m]
  t = time  [s]

Boundary conditions for carburising / nitriding:
  Initial condition:    C(x, 0) = C₀      for all x > 0
  Surface condition:   C(0, t) = C₄      for all t > 0
  Far-field condition: C(∞, t) = C₀     (core unaffected)

Semi-infinite slab solution (Crank, 1956):

  C(x,t) = C₄ − (C₄ − C₀) × erf(x / (2√(Dt)))

  Equivalently:
  C(x,t) = C₀ + (C₄ − C₀) × erfc(x / (2√(Dt)))

  where erf(z) = (2/√π) ∫₀ᵟ exp(−u²) du
        erfc(z) = 1 − erf(z)

Solving for Case Depth

To find the depth x at which the concentration equals a specified threshold C_th, rearrange the solution:

Set C(x,t) = Cₜₕ and solve for x:

  Cₜₕ = C₄ − (C₄ − C₀) × erf(z)   where z = x/(2√(Dt))

  erf(z) = (C₄ − Cₜₕ) / (C₄ − C₀)

  z = erfinv[(C₄ − Cₜₕ) / (C₄ − C₀)]

  x = 2 × z × √(D × t)

  Case depth (mm) = 2 × erfinv[(C₄ − Cₜₕ) / (C₄ − C₀)] × √(D × t) × 1000

Note: √(Dt) is the fundamental diffusion length parameter.
Doubling time quadruples D×t and multiplies √(Dt) by √2 ≈ 1.41.
Case depth scales with √t, not t (square-root time law).

Arrhenius Temperature Dependence of Diffusivity

Diffusivity is not a constant — it is a strongly temperature-dependent quantity that must be evaluated at the carburising or nitriding temperature before applying Fick’s law. The relationship follows the Arrhenius equation:

Arrhenius Diffusivity:

  D(T) = D₀ × exp(−Qₐ / (R × T))

Where:
  D₀  = pre-exponential factor  [m²/s]
  Qₐ  = activation energy for diffusion  [J/mol]
  R   = gas constant = 8.314 J/mol·K
  T   = absolute temperature  [K]

Carbon in austenite (γ-iron):
  D₀ = 2.0 × 10⁻⁵ m²/s
  Qₐ = 142,000 J/mol  (142 kJ/mol)

  At 920°C (1193 K):
    D = 2.0×10⁻⁵ × exp(−142000 / (8.314 × 1193))
      = 2.0×10⁻⁵ × exp(−14.31)
      = 2.0×10⁻⁵ × 6.06×10⁻⁷
      = 1.21×10⁻¹¹ m²/s  ≈ 1.2–1.7×10⁻¹¹ m²/s (literature range)

  At 960°C (1233 K):
    D = 2.0×10⁻⁵ × exp(−142000 / (8.314 × 1233))
      ≈ 3.5×10⁻¹¹ m²/s   (~2.1× faster than at 920°C)

Nitrogen in ferrite (α-iron) at nitriding temperatures:
  D₀ ≈ 3.0 × 10⁻⁷ m²/s
  Qₐ ≈ 77,000 J/mol  (77 kJ/mol)

  At 520°C (793 K):
    Dₙ ≈ 6 × 10⁻¹² m²/s

Effect of Temperature on Case Depth (Worked Comparison)

Carburising 8620 steel (C₀=0.18, C₄=0.95, Cₜₕ=0.35): t = 6 hr = 21,600 s

At 920°C: D = 1.65×10⁻¹¹ m²/s
  D×t = 1.65×10⁻¹¹ × 21600 = 3.564×10⁻⁷ m²
  √(Dt) = 5.97×10⁻⁴ m = 0.597 mm
  erf(z) = (0.95−0.35)/(0.95−0.18) = 0.60/0.77 = 0.7792
  z = erfinv(0.7792) ≈ 0.910
  ECD = 2 × 0.910 × 0.597 = 1.087 mm

At 950°C: D ≈ 2.8×10⁻¹¹ m²/s
  D×t = 2.8×10⁻¹¹ × 21600 = 6.048×10⁻⁷ m²
  √(Dt) = 7.78×10⁻⁴ m = 0.778 mm
  erf(z) = 0.7792 (same)  →  z = 0.910
  ECD = 2 × 0.910 × 0.778 = 1.416 mm

Conclusion: +30°C raises ECD from 1.09 mm to 1.42 mm (+30%)
at identical cycle time.

Case Hardening Processes: Carburising and Nitriding

Gas Carburising

Gas carburising is the most widely used case hardening process for steel gears, shafts, camshafts, and bearing races. The component is heated to the austenite region (900–980 °C) and held in a furnace atmosphere of endothermic gas (typically 40% N₂, 40% CO, 20% H₂) enriched with natural gas or propane to maintain a carbon potential of 0.8–1.1 wt%C at the steel surface. Carbon transfers from the gas atmosphere to the austenite by the reaction:

The carbon potential is controlled by monitoring the dew point, CO₂ content, or oxygen sensor (Zirconia probe) of the atmosphere and adjusting the enrichment gas flow rate through a PID controller. After the carburising soak, the component is either direct-quenched from the furnace temperature or slow-cooled and subsequently re-austenitised and quenched. See the case hardening processes guide for detailed atmosphere control and process monitoring.

Vacuum (Low-Pressure) Carburising

LPC operates at 1–10 mbar using acetylene (C₂H₂) or propane pulsed into the furnace chamber. The process alternates between carburise pulses (gas on: 5–60 s) and diffusion pauses (gas off: surface carbon redistributes inward). Multiple boost-diffuse cycles build the target profile. Because no oxygen is present, intergranular oxidation is entirely eliminated. Higher operating temperatures (up to 1050 °C with grain-refined steels) dramatically accelerate the process. The Fick’s law model governs the diffusion steps; process simulation software (e.g. SimCarb, DANTE) iterates the profile through successive boost-diffuse sequences to design the cycle recipe.

Gas Nitriding

Gas nitriding exposes components to a dissociated ammonia atmosphere (NH₃ ⇒ N + 3/2 H₂) at 500–580 °C in the ferritic field. Nitrogen dissociating at the surface diffuses into the steel, forming a compound layer (white layer: γ’-Fe₄N and/or ε-Fe_2-3N, 5–25 μm thick) and a deeper diffusion zone (nitrogen in solid solution and fine nitride precipitates, 0.1–0.5 mm deep). Because the process occurs below A_c1, no phase transformation occurs and no post-nitriding hardening is required — the component is already in its final tempered condition. Core hardness is fully retained. Surface hardness depends on alloy content: plain carbon steel achieves 250–300 HV; steels containing Cr, Mo, Al, V achieve 700–1200 HV from alloy nitride precipitation.

Ferritic Nitrocarburising (FNC)

FNC (also marketed as Tenifer, Tufftride, Melonite) simultaneously introduces both carbon and nitrogen at 540–590 °C. The compound layer formed is predominantly ε-Fe_2-3(N,C), which has superior ductility compared with the brittle ε-nitride layer. FNC is widely applied to crankshafts, camshafts, piston rods, and tools where wear resistance, fatigue strength improvement, and moderate corrosion resistance are required without dimensional change. The diffusion zone depth is predicted using the nitrogen diffusivity in ferrite at the FNC temperature.

Industrial Applications and Case Depth Specifications

Gear Tooth Carburising

Case depth specification for carburised and case-hardened gears follows ISO 6336-5 and AGMA 2001. Effective case depth (ECD) at 550 HV (approximately 0.35 wt%C threshold after hardening) is specified as a function of gear module (tooth size). The general guideline is ECD ≈ 0.15–0.20 × module for spur gears, ensuring that the case extends to at least 30% of the tooth depth. Core hardness after quench and temper is typically 25–40 HRC for Ni-Cr-Mo steels (SAE 8620, 9310, EN 36) — providing the tough, impact-resistant core required for heavy-duty transmission applications. The martensite formation article explains the role of carbon content in controlling case and core hardness.

Bearing Races and Rollers

Bearing components require very fine microstructure (ASTM grain size 7–10) and precise retained austenite control (typically 5–20% RA after subzero treatment at −60 to −80 °C) to prevent dimensional instability in service. Carbon content in the finished case must reach 0.75–0.95 wt%C to achieve full case hardness. Vacuum carburising at 930–980 °C with nitrogen quenching (high-pressure gas quench) provides superior uniformity and eliminates the soft spots associated with liquid quenching flow irregularities. SAE 9310 and M50 Nil (Ni-Mo-Cr alloy bearing steel) are standard aerospace bearing grades processed by LPC.

Nitrided Crankshafts and Camshafts

Automotive crankshafts in ductile iron and medium-carbon alloy steels (42CrMo4, 40CrNiMo) are gas nitrided or FNC treated to develop a compound layer (2–15 μm ε-Fe_2-3(N,C)) on all journal surfaces, improving wear resistance and fatigue strength by introducing beneficial compressive residual stresses. The diffusion zone (0.1–0.35 mm) provides the fatigue endurance improvement. No dimensional distortion occurs because the process temperature is well below the prior tempering temperature. White layer thickness is controlled through nitriding potential (K_N = p_NH3/p_H2^1.5) and temperature; an excessively thick or brittle white layer is detrimental and is specified at 10–25 μm maximum for most automotive applications.