Metallurgical Heat Treatment, Gear Gas/Vacuum Carburizing & Fick’s Second Law
Steel Gas Carburizing Harris Case-Depth & Time Formula
Calculate Arrhenius Carburizing Diffusion Factor KT (mm/sqrt(hr)), Required Furnace Soak Hours (t = (ECD/KT)²), Case Depth Achieved at 4 Hours (mm), and Total Cycle Minutes.
Doubling carburized gear case depth takes FOUR TIMES the furnace hours—unless austenite temperature is boosted!
Interstitial carbon diffusion into low-carbon alloy gear steel (`8620`, `18CrNiMo7-6`) held in endothermic or low-pressure acetylene austenite (`870–980 °C`) obeys Adolf Fick’s Second Law `dC/dt = D * d²C/dx²`, whose complementary error-function solution scales strictly with the **square root of time**! F. E. Harris (1944) calibrated the universal shop-floor Arrhenius Case-Depth equation (converted to SI millimeters with `T_K = T_C + 273.15`): `ECD_mm = K_T * SQRT(t_hr)` where `K_T = 802.6 * EXP(-8,588 / T_K)` (`mm / SQRT(hr)`)! Because `t_hr = (ECD_mm / K_T)^2`, doubling case depth from `0.60` mm to `1.20` mm at constant furnace temperature quadruples soak time `4x`—whereas raising low-pressure vacuum carburizing temperature by just `+30 °C` exponentially accelerates `K_T` and cuts furnace hours nearly `30%`!
Verify on an 8620 Pinion-Gear Endothermic Gas Carburizing Cycle—Furnace_Austenite_Temp `T` = 925.0 °C (`1,198.15` K, B2 -> Harris Rate Factor `K_T = 0.619` mm/√hr in E2) and Target_Case_Depth `ECD` = 1.20 mm (B3): Required Carburizing Time equals `(1.20 / 0.619)^2 = 3.76` hours (E3 = `225.6` minutes in E5), while a standard `4.0`-hour reference soak produces `1.24` mm case depth (E4)!
| Harris Fickian Carburizing KPI (925 °C Austenite, Target ECD = 1.20 mm) | Formula | Verified Output |
|---|---|---|
| Arrhenius Factor KT (mm/√hr) | 802.6 * EXP(-8588 / (Temp_C + 273.15)) | 0.619 mm/√hr (0.737 @ 955 °C vacuum!) |
| Required Furnace Soak (hours) | (Target_ECD_mm / KT_Factor) ^ 2 | 3.76 hr (2.65 hr = -29.5% @ 955 °C!) |
| Reference 4-Hour Case Depth (mm) | KT_Factor * SQRT(4) | 1.24 mm (1.47 mm @ 955 °C) |
| Required Carburizing Cycle (min) | Required_Soak_Hours * 60 | 225.6 min (158.9 min @ 955 °C) |
Harris Arrhenius Root-Time Carburizing Case-Depth & Soak Duration Formula
Validate Austenite Carburizing `Temp_C` in `[840, 1,050]` °C (above `Ac3` austenite transformation and below grain-coarsening incipient fusion!) and `Target_ECD` in `(0.05, 6.0]` mm!
=LET(tc_c, B2, ec_m, B3, IF(OR(tc_c < 840, tc_c > 1050, ec_m <= 0.05, ec_m > 6), "Invalid Carburizing Input", LET(kt_f, 802.6 * EXP(-8588 / (tc_c + 273.15)), hr_s, (ec_m / kt_f) ^ 2, ROUND(hr_s * 60, 1))))955 °C Low-Pressure Vacuum Carburizing Temperature Boost Check (Set Temp B2 = 955.0 °C)
Step furnace temperature up `+30 °C` to `955.0 °C` (`B2 = 955`) in a micro-alloyed fine-grain gear steel: Arrhenius Factor `K_T` E2 jumps from `0.619` to `0.737` mm/√hr, slashing Required Furnace Time E3 from `3.76` hr down to `2.65` hr (`158.9` min in E5—saving over `66` minutes per furnace load)!
=IF(ISTEXT(E5), E5, LET(kt_f, 802.6 * EXP(-8588 / (B2 + 273.15)), ROUND((B3 / kt_f) ^ 2, 2)))Try the example
Use the original values or change them to test your understanding.