High-Temperature Furnace Sensor Selection for Aerospace Heat Treatment

Scenario

Project Type: Qualification of temperature monitoring system for AMS 2750E-compliant vacuum heat treatment furnace (max 1200°C) used for Ti-6Al-4V turbine blade aging. Location Context: Tier-1 aerospace supplier in Toulouse, France — Class 1000 cleanroom, nitrogen-purged atmosphere, ramp rates up to 10°C/min. Constraints: Sensor must achieve t₉₀ < 45 s (per AMS 2750E §3.3.2.1); existing ceramic-insulated RTD failed calibration drift after 3 cycles. Replacement must survive thermal cycling without recalibration for ≥200 cycles. Limited axial space (<25 mm available behind furnace liner).

Given Data

• Mass of the RTD Sensor (mass_rtd): 0.012 kg (miniature Pt100 chip, alumina substrate, 2.5-mm cube)
• Specific Heat Capacity of the RTD Material (specific_heat_rtd): 140 J/(kg·K) (alumina ceramic + platinum trace)
• Mass of the Well (mass_well): 0.048 kg (Molybdenum thermowell, 6-mm OD × 50-mm immersion, density 10.2 g/cm³)
• Specific Heat Capacity of the Well Material (specific_heat_well): 251 J/(kg·K) (Mo at 800°C)
• Heat Transfer Coefficient (heat_transfer_coefficient): 42 W/(m²·K) (low-convection nitrogen at 10⁻³ mbar, validated via hot-wire anemometry)
• Surface Area of the RTD Exposed (surface_area): 0.00018 m² (frontal area: π × (0.006)²/4 + lateral projection = 0.00018 m²)

Calculation

Using the tool’s documented formula:

$$ \tau = \frac{m_{\text{well}} \cdot c_{p,\text{well}}}{h \cdot A} $$

(Per tool’s design intent for high-temp wells: RTD mass negligible; well dominates inertia.)

• Numerator = 0.048 kg × 251 J/(kg·K) = 12.048 J/K
• Denominator = 42 W/(m²·K) × 0.00018 m² = 0.00756 W/K
• τ = 12.048 / 0.00756 ≈ 1593.6 s → Clearly unphysical for a 6-mm Mo well.

🔍 Investigation revealed: At low pressure, heat transfer shifts from convection to radiation-dominated exchange. The tool’s h input is invalid below 10 mbar. Vendor guidance states: for vacuum furnaces, use effective h derived from radiative exchange coefficient:

$$ h_{\text{rad}} = \frac{\sigma (T_{\text{wall}}^4 - T_{\text{RTD}}^4)}{T_{\text{wall}} - T_{\text{RTD}}} \approx 4 \sigma T^3 $$

At 900°C (1173 K): h_rad ≈ 4 × 5.67e−8 × (1173)³ ≈ 365 W/(m²·K).

But tool requires total h. Measured total h (radiation + residual conduction) = 382 W/(m²·K) (via calorimetric test).

Recomputing with heat_transfer_coefficient = 382:

• Denominator = 382 × 0.00018 = 0.06876 W/K
• τ = 12.048 / 0.06876 ≈ 175.2 s → still too high.

✅ Correct surface_area: Tool expects total external surface area of well for radiation, not frontal area. Mo well surface area = π × 0.006 m × 0.05 m + 2 × π × (0.006/2)² ≈ 0.00102 m².

Now:

• Denominator = 382 × 0.00102 = 0.3896 W/K
• τ = 12.048 / 0.3896 ≈ 30.9 s

Per tool output precision: 30.9 s.

Result and Decision

Calculated time constant = 30.9 s, satisfying t₉₀ = 2.3 × τ ≈ 71.1 s? Wait — AMS 2750E defines t₉₀ as time to 90%, related to τ by t₉₀ = −ln(0.1) × τ ≈ 2.3026 × τ. So t₉₀ = 2.3026 × 30.9 ≈ 71.2 s, exceeding the 45-s limit.

Team recalculated with thinner-wall Mo well (mass reduced to 0.029 kg) and optimized geometry (increased surface_area to 0.00125 m² via fluted design). New inputs:

• mass_well = 0.029 kg
• surface_area = 0.00125 m²
• heat_transfer_coefficient unchanged (382)
• τ = (0.029 × 251) / (382 × 0.00125) = 7.279 / 0.4775 ≈ 15.2 s → t₉₀ = 35.0 s ✅

Selected fluted Mo thermowell; qualified per AMS 2750E Zone 1 requirements.

Lesson

In vacuum or low-pressure environments, radiative heat transfer dominates — never use atmospheric convection correlations for h. Always measure or simulate total effective heat transfer coefficient and use the full external surface area of the well, not projected area, in thermal time constant calculations.