What Is the Quantitative Thermal Stress (MPa) Developed in a 2.5 mm Thick Titanium Heater Wall During a Quench from 95°C to 20°C Water?

A process engineer concerned about thermal shock damage to a thick-walled titanium heater (2.5 mm) after quenching from 95°C to 20°C water can determine the quantitative thermal stress created in the wall from the thermal stress equation for a rapidly cooled tube. Maximum thermal stress (σ th ) occurs at the inner surface (assuming rapid cooling from the outside) and is given by: σ th = E ×α × T Δ 1 − v f(β) Where E is elastic modulus (110 GPa), α is coefficient of thermal expansion (8.6 × 10 -6 /°C), T Δ is the temperature change (75°C), ν is Poisson's ratio (0.34) and f(β) is a geometry factor dependent on the Biot number (cooling rate). For a thick-walled tube (t = 2.5 mm, OD = 25 mm) quenched in water (h = 1,000-5,000 W/m²K, Biot number > 10) f(β) ≈ 1.0 (complete thermal shock). The computed thermal stress is σ_th = (110 × 10 9 × 8.6 × 10 6 × 75) / (1 − 0.34) = (110e9 × 6.45e-4) / 0.66 = (71 × 10 6) / 0.66 = 108 MPa. This stress is around 39 percent of the yield strength (275 MPa) and 54 percent of the maximum tensile strength (345 MPa). Thermal fatigue cracking can be induced by repeated thermal shocks (100-1,000 cycles) at stresses below the yield strength.

Mechanism of Formation of Thermal Stresses During Quenching
If a hot tube of titanium is plunged into cold water the outside surface cools and compresses rapidly. The inside surface, remaining hot, opposes this contraction, putting the outside surface in tension and the inside surface in compression (for a tube cooled from the outside). For high heat transfer coefficients (h > 5,000 W/m²K) the temperature difference across the wall thickness can be 30-50°C for a 2.5 mm wall during the first few seconds of quenching. The thermal stress is related to the instantaneous temperature difference across the wall, and it is maximum at about 0.3-0.5 $\times$ $\Delta T_{total}$. The calculated 108 MPa is the maximum stress during the quench. If the yield strength is surpassed, plastic deformation takes place and residual tensile stresses are present after cooling, increasing the susceptibility to stress corrosion cracking.

Quantification of Thermal Stress as a Function of Wall Thickness and Quench Severity
Wall Thickness t (mm) Quench Medium Heat Transfer Coefficient h (W/m 2 K) Max ΔT Across Wall (°C) Thermal Stress (MPa) % of Yield (275 MPa)Plastic Deformation Hazard
1.5 Still water (mild) 500 15 22 8%None 1.5 Agitated water 2,000 30 43 16% None
1.5 Quench (cold water) 5,000 45 65 24% None
2.0 Quench (cold water) 5,000 55 79 29% None 2.5 Quench (cold water) 5,000 65 108 39% None (elastic)
3.0 Quench (Cold water) 5,000 75 125 45% None (Elastic)
2.5 Ice water (0°C, ΔT=95°) 10,000 85 140 51% Close to yield
3.0 Ice water (0°C) 10000 100 165 60%Plastic deformation is likely
A Scenario-Based Guide to Thermal Shock Management
Operating Condition & Quench Type Wall Thickness (mm) Thermal Stress (MPa) Cracking RiskRecommended Action:
Normal shutdown, air cool (slow) 1.5-3.0 < 10 NothingNo problems
Normal Shutdown, Water Rinse (Warm) 1.5-2.0 20-40 None Acceptable
Emergency quench (cold water, one-time) 2.5 108 Low (one-time) Acceptable. Post event check.
Emergency Quench (ice water, single incident) 2.5 140 Moderate (may yield locally)Remove heater for examination Warp check.
2.5 108 Moderate (fatigue) Increase quench temperature (warm water) Repeated thermal cycles (> 100) with water quench Reduced cooling rate.
Frequent (daily) quenchingAny > 2.0 > 100 High (fatigue cracking) Use ramp cooling (controlled rate) instead of quench.
Quench from high temperature (> 150°C) 2.5 > 200 SevereNot advised. Use slow cooling or other substance.
Engineering mitigation of quench-induced thermal stress
Thermal quenching is unavoidable on thick-walled titanium heaters but there are three mitigations that help reduce thermal stress. The first mitigation is the use of warm quench water (40-50 °C vs. 20 °C). Reducing ΔT from 75°C to 45°C reduces thermal stress from 108 MPa to 65 MPa (40% reduction), stress far below yield. The second mitigation is to use still water instead of agitated water (h=500 W/m2K instead of 5,000) to increase the quench time. The reduced heat transfer coefficient reduces the temperature gradient through the wall, reducing the peak stress to 22 MPa for a 2.5 mm wall. The third mitigation is pre-cooling of the heater in air before water quenching. Cooling in air from 95°C to 60°C (10 min) followed by quenching reduces ΔT from 75°C to 40°C and hence thermal stress.

Conclusion: 2.5 mm Titanium Wall Sees 108 MPa Thermal Stress During 95 C -> 20 C Quench
A 2.5 mm thick Grade 2 titanium heater wall experiences a quantitative thermal stress of about 108 MPa during a quench from 95°C to 20°C water with a heat transmission coefficient of 5,000 W/m2K (agitated water). This stress is 39% of the material yield strength (275 MPa) and 54% of the maximum tensile strength (345 MPa). The tension is still in the elastic range such that a single quench will not generate plastic deformation or rapid failure. Repeated heat quenches (100-1,000 cycles) can induce thermal fatigue cracking, particularly at welds, scratches or other stress concentrators. Warm quench water (40-50 °C), moderate cooling in still water and air pre-cooling before quenching are recommended to improve heater life. Where frequent quenching is needed, lowering the wall thickness to 1.5-2.0 mm reduces the thermal stress to 65-79 MPa, well within the safe elastic range.