Temperature Sensitivity Assessment

Reference ID: MET-A157 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The temperature-sensitivity assessment determines whether a heat-generating component can be kept within its allowable case temperature when exposed to forced-air or liquid cooling. In process engineering this analysis is essential for sizing heat-removal hardware, ensuring reliability of electronic modules, and complying with safety limits in high-power equipment such as power supplies, motor drives, and process control units.

The calculation evaluates the convective heat-transfer coefficient for forced air using an empirical correlation, checks the applicability of the lumped-capacitance model via the Biot number, and compares the predicted temperature rise for air versus a prescribed liquid-cooling coefficient. The results guide the selection of cooling strategy or the redesign of the thermal interface.

Methodology & Formulas

The analysis follows a deterministic sequence of physics-based steps. All symbols are defined in the code block and are used here algebraically.

1. Reynolds number for the air flow

\[ Re_{\text{air}} \;=\; \frac{V_{\text{air}}\,L_{c}}{\nu_{\text{air}}} \]

2. Validity range for the forced-air correlation

Parameter	Valid Range
Reynolds number \(Re_{\text{air}}\)	\(5\times10^{3} \le Re_{\text{air}} \le 1\times10^{5}\)

3. Convective heat-transfer coefficient for forced air

The explicit correlation combines the Reynolds and Prandtl numbers: \[ h_{\text{air}} \;=\; C_{\text{air}}\, \left(Re_{\text{air}}\right)^{m_{\text{air}}}\, \left(Pr_{\text{air}}\right)^{n_{\text{air}}}\, \frac{k_{\text{air}}}{L_{c}} \]

4. Biot number – lumped-capacitance applicability

\[ Bi \;=\; \frac{h_{\text{air}}\,L_{c}}{k_{s}} \]

Criterion	Threshold
Biot number	\(Bi < 0.1\) (lumped capacitance valid)

5. Temperature budget

Maximum allowable temperature rise: \[ \Delta T_{\max} \;=\; T_{\text{limit}} - T_{\text{amb}} \]

6. Predicted temperature rise with air cooling

\[ \Delta T_{\text{air}} \;=\; \frac{Q}{h_{\text{air}}\,A} \]

7. Predicted temperature rise with liquid cooling (using an estimated coefficient)

\[ \Delta T_{\text{liq}} \;=\; \frac{Q}{h_{\text{liq,est}}\,A} \]

8. Required inlet temperature of the coolant

To keep the case temperature at or below the limit when liquid cooling is employed: \[ T_{\text{cool,in}} \;=\; T_{\text{limit}} - \Delta T_{\text{liq}} \]

9. Summary of decision criteria

Check	Condition	Interpretation
Reynolds number range	\(Re_{\text{air}} < 5\times10^{3}\) or \(Re_{\text{air}} > 1\times10^{5}\)	Correlation may be inaccurate – reconsider flow regime or use a different model.
Biot number	\(Bi \ge 0.1\)	Lumped-capacitance assumption questionable – a distributed thermal model may be required.
Air-cooling temperature rise	\(\Delta T_{\text{air}} > \Delta T_{\max}\)	Air cooling insufficient – consider higher flow rate, larger surface area, or liquid cooling.

A Temperature Sensitivity Assessment evaluates how variations in temperature affect process performance, product quality, and safety. Key reasons it is critical:

Identifies optimal operating windows to maximize yield and minimize energy consumption.
Highlights temperature-driven safety risks such as runaway reactions or equipment stress.
Supports robust control strategy development by quantifying allowable temperature deviations.
Enables compliance with regulatory specifications that often include temperature limits.

Follow a systematic approach:

Literature Review: Gather reported operating temperatures for similar chemistries.
Thermodynamic Screening: Use equilibrium calculations to estimate feasible temperature limits.
Laboratory Kinetic Trials: Conduct small-scale experiments across a broad temperature span to map conversion and selectivity trends.
Safety Margin Assessment: Apply safety factors for exothermicity, pressure rise, and equipment material limits.
Scale-up Validation: Verify the identified window in pilot-scale runs before finalizing the design setpoint.

Commonly used techniques include:

Regression Analysis: Fit temperature versus response (e.g., conversion) to quantify sensitivity coefficients.
Analysis of Variance (ANOVA): Determine if temperature effects are statistically significant compared to other factors.
Design of Experiments (DoE): Employ factorial or response-surface designs to capture interaction effects.
Monte Carlo Simulation: Propagate temperature variability through the model to predict distribution of outcomes.
Control Charting: Monitor process temperature trends over time for early detection of drift.

Effective documentation and communication involve:

Executive Summary: One-page overview of findings, recommended temperature window, and business impact.
Technical Report: Detailed methodology, data tables, statistical analysis, and safety considerations.
Visual Aids: Graphs such as temperature-vs-conversion curves, contour plots, and risk heat maps.
Actionable Recommendations: Clear setpoints, control tolerances, and contingency plans.
Stakeholder Briefings: Tailor presentations for engineering, operations, quality, and management audiences, highlighting relevance to each group.

Worked Example – Checking Whether a Small Heat Sink Will Stay Below 80 °C

A process skid is being designed to cool a 150 W power module with a 20 cm² aluminum heat sink. The skid will be located in a 25 °C control room and the module must not exceed 80 °C. The designer needs to know if the heat sink can keep the module below the limit when the room air is moving at 5 m s⁻¹.

Knowns

Heat load, \(Q\) = 150 W
Maximum allowable module temperature, \(T_{\text{limit}}\) = 80 °C
Ambient (room) temperature, \(T_{\text{amb}}\) = 25 °C
Heat-sink planform area, \(A\) = 0.02 m²
Characteristic length (fin spacing), \(L_c\) = 0.015 m
Air velocity, \(V_{\text{air}}\) = 5 m s⁻¹
Air properties at 25 °C: \(\rho_{\text{air}}\) = 1.204 kg m⁻³, \(\nu_{\text{air}}\) = 1.5 × 10⁻⁵ m² s⁻¹, \(k_{\text{air}}\) = 0.0257 W m⁻¹ K⁻¹, \(\Pr_{\text{air}}\) = 0.71
Heat-sink material conductivity, \(k_s\) = 200 W m⁻¹ K⁻¹
Empirical constants for forced convection: \(C_{\text{air}}\) = 0.026, \(m_{\text{air}}\) = 0.8, \(n_{\text{air}}\) = 0.33

Step-by-step calculation

Compute the Reynolds number based on \(L_c\): \[ \Re_{\text{air}} = \frac{V_{\text{air}} L_c}{\nu_{\text{air}}} = \frac{5 \times 0.015}{1.5 \times 10^{-5}} = 5000 \]
Evaluate the average air-side heat-transfer coefficient using the correlation for forced flow over a plate: \[ h_{\text{air}} = \frac{C_{\text{air}} k_{\text{air}}}{L_c} \Re_{\text{air}}^{m_{\text{air}}} \Pr_{\text{air}}^{n_{\text{air}}} = \frac{0.026 \times 0.0257}{0.015} \times 5000^{0.8} \times 0.71^{0.33} = 36.2\ \text{W m}^{-2}\text{ K}^{-1} \]
Check the Biot number to see if the heat-sink base can be treated as nearly isothermal: \[ \text{Bi} = \frac{h_{\text{air}} L_c}{k_s} = \frac{36.2 \times 0.015}{200} = 0.003 \] Because Bi ≪ 0.1, the temperature gradient inside the base is negligible.
Calculate the maximum allowable temperature rise above ambient: \[ \Delta T_{\text{max}} = T_{\text{limit}} - T_{\text{amb}} = 80 - 25 = 55\ \text{K} \]
Compute the actual temperature rise of the heat-sink surface: \[ \Delta T_{\text{air}} = \frac{Q}{h_{\text{air}} A} = \frac{150}{36.2 \times 0.02} = 207\ \text{K} \]

Final Answer

The required temperature rise is 55 K, but the heat sink can only sustain 207 K above ambient under the specified air flow. Therefore, the 150 W module will exceed the 80 °C limit; additional surface area, higher air speed, or liquid cooling is required.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle