Technical Ceramics, Thermoelasticity & High-Temperature Materials
Ceramic Thermal-Shock Resistance R & R-Prime Formula
Calculate Hasselman/Kingery Severe-Quench Critical Delta-T Parameter R (K), Conduction-Weighted Parameter R-Prime (W/m), Biaxial Quench Surface Stress (MPa), and Thermal-Shock Safety Factor.
High stiffness E and high expansion alpha shatter strong ceramics—R = sigma_f*(1-nu)/(E*alpha) rules quenches!
When a red-hot ceramic component plunges into cold water (Biot Number Bi >> 1), its skin contracts instantaneously while its hot core resists contraction, generating biaxial surface tensile stress sigma_th = (E * alpha * Delta_T) / (1 - nu)! Setting sigma_th = Flexural_Fracture_Strength sigma_f defines Kingery & Hasselman’s First Thermal-Shock Resistance Parameter R = (sigma_f * (1 - nu)) / (E * alpha) (critical instantaneous quench Delta_T_crit in K), while mild convective cooling (Bi < 1) scales with the Second Parameter R_prime = R * k (W/m)! Note unit alignment: E_GPa * 1000 * alpha_ppm_K * 10^-6 = E_GPa * alpha_ppm_K / 1000 MPa/K!
Verify on Silicon Nitride (Si3N4) Turbine Hardware—Flexural_Strength = 720 MPa (B2), Poisson_Ratio nu = 0.25 (B3), Young_Modulus E = 300 GPa (B4), Thermal_Expansion_CTE alpha = 3.0 ppm/K (B5), Thermal_Conductivity k = 30 W/(m·K) (B6), and Quench_DeltaT = 400 K (B7): Critical Quench Parameter R reaches (720 * 0.75) / (300 * 3.0 / 1000) = 600.00 K (E2), Conduction Figure R_prime hits 18,000.00 W/m (E3), Actual Surface Tensile Stress is 480.00 MPa (E4), and Safety Factor is 600 / 400 = 1.50x (E5)!
| Hasselman Ceramic Quench KPI (Si3N4: 720 MPa, nu=0.25, 300 GPa, 3.0 ppm/K) | Formula | Verified Output |
|---|---|---|
| Critical Quench Delta-T R (K) | sigma_f * (1 - nu) / (E_GPa * alpha_ppm / 1000) | 600.00 K (95.00 K Alumina) |
| Mild-Quench Parameter R' (W/m) | Critical_R_K * Conductivity_W_mK | 18,000.00 W/m (2,850.00 Al2O3) |
| Biaxial Surface Stress (MPa) | (E_GPa * alpha_ppm / 1000) * dT / (1 - nu) | 480.00 MPa (1,600.00 Al2O3!) |
| Thermal-Shock Safety Factor | Critical_R_K / Actual_Quench_dT_K | 1.50x (0.24x Shatter!) |
Kingery-Hasselman Ceramic Critical Quench Delta-T Formula
Validate Flexural_Strength > 0, Poisson_Ratio in (0, 0.50), Young_Modulus_GPa > 0, CTE_ppm_K > 0, Conductivity > 0, and Quench_DeltaT > 0!
=LET(sg_f, B2, nu_p, B3, md_g, B4, al_p, B5, kc_w, B6, dt_q, B7, IF(OR(sg_f <= 0, nu_p <= 0, nu_p >= 0.5, md_g <= 0, al_p <= 0, kc_w <= 0, dt_q <= 0), "Invalid Ceramic Quench Input", ROUND((sg_f * (1 - nu_p)) / (md_g * al_p / 1000), 2)))Sintered Alumina (Al2O3) Thermal-Shock Shatter Check (380 MPa, nu=0.24, 380 GPa, 8.0 ppm/K)
Substitute dense Alumina Al2O3 properties (B2 = 380 MPa, B3 = 0.24, B4 = 380 GPa, B5 = 8.0 ppm/K): Critical Quench R E2 plunges from 600.00 K to just 95.00 K, Biaxial Quench Stress at 400 K E4 spikes to 1,600.00 MPa, and Safety Factor E5 collapses to 0.24x—explaining why alumina crucibles crack immediately on water quenching!
=IF(ISTEXT(E2), E2, ROUND(E2 / B7, 2))Try the example
Use the original values or change them to test your understanding.