import marimo __generated_with = "0.25.1" app = marimo.App() @app.cell def _(): import marimo as mo return (mo,) @app.cell(hide_code=True) def _(mo): mo.md(r""" # Full Uncertainty Budget of a Hot-Wire Anemometer for Isothermal Turbulent Air Flow in a Wind Tunnel ## Overview For isothermal turbulent air flow measurements in a wind tunnel where air temperature increases with time due to friction, the uncertainty budget of a hot-wire anemometer becomes significantly more complex than for truly isothermal conditions. The combined standard uncertainty typically ranges from **8-15%**, with an expanded uncertainty (k=2, 95% confidence) of **16-30%** depending on the specific experimental conditions and mitigation strategies employed.[^1][^2][^3] ## Major Uncertainty Categories ### 1. Calibration Uncertainties (Combined: ~3.9%) **Reference Velocity Measurement (0.2-0.5%)**[^4][^1] - Static pressure measurement uncertainty from Pitot tube - Dynamic pressure transducer accuracy - Flow uniformity in calibration facility **Reference Temperature Measurement (0.5-1.0%)**[^5][^6] - Thermocouple or RTD sensor accuracy - Temperature sensor positioning relative to hot-wire - Ambient temperature monitoring during calibration **Calibration Curve Fitting (0.5-2.0%)**[^7][^2] - Polynomial fitting residuals for voltage-velocity relationship - King's law parameter determination accuracy - Calibration point distribution and range adequacy **Calibration Drift (1.0-5.0%)**[^8][^9] - Time-dependent changes in wire characteristics - Contamination and wire aging effects - Changes in ambient conditions between calibration and measurement **Temperature Compensation Errors (1.0-3.0%)**[^10][^11][^8] - Accuracy of temperature correction models - Validity of heat transfer correlations - Assumption of constant wire overheat ratio ### 2. Instrumentation Uncertainties (Combined: ~1.8%) **Hot-wire Sensor Characteristics (0.5-2.0%)**[^7][^5] - Wire material property variations - Geometric tolerances (diameter, length) - Wire mounting and support effects - Wire uniformity along its length **Wire Positioning and Alignment (0.5-1.5%)**[^1][^5] - Alignment with flow direction - Position accuracy in the measurement volume - Probe traversing system accuracy - Effects of probe support interference **CTA Electronics (0.3-1.0%)**[^12][^1] - Bridge stability and temperature coefficient - Amplifier gain and offset stability - Power supply regulation - Electronic drift over measurement duration **Analog-to-Digital Conversion (0.02-0.1%)**[^12][^1] - Quantization error (typically 16-bit resolution) - Input range optimization - Anti-aliasing filter characteristics ### 3. Environmental and Flow Uncertainties (Combined: ~5.0%) **Ambient Temperature Drift (2.0-5.0%)**[^13][^8][^10] - Most critical uncertainty for non-isothermal flows - Diurnal temperature variations - Laboratory climate control stability - Heat sources in the facility affecting local temperature **Temperature Rise Due to Friction (1.0-5.0%)** - Viscous dissipation in turbulent boundary layers - Heat generation from fan/blower operation - Flow acceleration and compression effects - Inadequate heat removal from the wind tunnel circuit **Air Density Variations (0.5-2.0%)**[^11][^14] - Pressure and temperature dependence of air properties - Altitude and weather-related barometric pressure changes - Local pressure variations due to wind tunnel operation **Humidity Effects (0.5-2.0%)**[^14] - Changes in air thermal conductivity with water vapor content - Seasonal and daily humidity variations - Condensation effects on wire surface ### 4. Spatial and Temporal Resolution Limitations (Combined: ~6.9%) **Wire Length Effects (1.0-5.0%)**[^15][^16][^17] - Spatial averaging over the wire length relative to turbulence length scales - Under-resolution of small-scale turbulent structures - Viscous-scaled wire length (l+) considerations - Wire length to diameter ratio effects **Frequency Response Limitations (2.0-10.0%)**[^18][^19][^20] - CTA bandwidth limitations at high frequencies - System tuning and stability trade-offs - Wire thermal time constant effects - Electronic circuit response characteristics **Wire Thermal Inertia (0.5-2.0%)**[^15][^18] - Wire thermal time constant limitations - Material property effects on temporal response - Wire diameter and length optimization trade-offs **Probe Interference Effects (0.5-2.0%)**[^21][^16] - Flow disturbance from probe supports and prongs - Wake effects downstream of the probe - Blockage effects in confined flows ### 5. Data Processing and Analysis Uncertainties (Combined: ~6.3%) **Non-isothermal Flow Corrections (2.0-10.0%)**[^22][^23][^10] - Accuracy of temperature compensation algorithms - Validity of correction models for varying temperature - Real-time temperature measurement and correction **Statistical Sampling Errors (0.5-2.0%)**[^24][^25] - Finite sampling time effects on turbulence statistics - Convergence of statistical moments - Sample size adequacy for desired confidence levels **Turbulence Corrections (0.5-2.0%)**[^16][^17] - Corrections for spatial resolution effects - High-frequency attenuation corrections - Turbulence intensity-dependent corrections ## Critical Considerations for Non-Isothermal Conditions ### Temperature Drift Compensation The most significant challenge in "isothermal" turbulent flow with temperature rise due to friction is the **temperature drift compensation**. Several approaches can be employed:[^8][^10] 1. **Intermediate Single Point Recalibration (ISPR)** - Periodic recalibration at a reference point[^8] 2. **Multi-temperature calibration** - Calibration curves at different temperatures[^10][^11] 3. **Real-time temperature correction** - Continuous temperature monitoring and correction[^22] 4. **Power-to-Resistance Ratio (PDR) method** - Temperature-independent velocity measurement[^22] ### Measurement Protocol Recommendations To minimize uncertainties in non-isothermal conditions: 1. **Pre- and post-calibration** with temperature monitoring[^15][^8] 2. **Temperature-controlled test sections** where feasible 3. **Rapid data acquisition** to minimize exposure to temperature drift 4. **Multiple measurement repetitions** with statistical analysis 5. **Temperature logging** throughout the measurement campaign ## Combined Uncertainty Assessment Using the root-sum-of-squares method for independent uncertainty components, the typical combined standard uncertainty is approximately **11.4%**, resulting in an expanded uncertainty of **22.9%** at 95% confidence (k=2). The dominant contributors are: 1. **Frequency response limitations** (6.0%) 2. **Non-isothermal corrections** (6.0%) 3. **Temperature drift** (3.5%) 4. **Calibration drift** (3.0%) 5. **Temperature rise due to friction** (3.0%) Temperature-related effects contribute approximately **7.8%** to the combined uncertainty, while other effects contribute **8.3%**. This demonstrates that **temperature effects dominate the uncertainty budget** for non-isothermal turbulent flow measurements. ## Uncertainty Reduction Strategies 1. **Enhanced temperature control** in the wind tunnel circuit 2. **Improved calibration procedures** with temperature compensation 3. **Higher-frequency CTA systems** to reduce temporal resolution errors 4. **Shorter wire lengths** to improve spatial resolution 5. **Real-time correction algorithms** for temperature variations 6. **Statistical validation** through replicate measurements The uncertainty budget should be evaluated according to GUM (Guide to the Expression of Uncertainty in Measurement) principles, with proper classification of Type A (statistical) and Type B (other) uncertainty components, and appropriate propagation of uncertainties through the measurement equation.[^26][^27]
⁂
[^1]: https://turbmodels.larc.nasa.gov/Other_exp_Data/Notredame_sep/Documentation/Hot-Wire-Documentation-V2.pdf [^2]: https://www.imeko.info/publications/wc-2006/PWC-2006-TC21-021u.pdf [^3]: https://www.sciencedirect.com/science/article/pii/S099775461730496X [^4]: https://research.engineering.ucdavis.edu/wind/wp-content/uploads/sites/17/2014/03/AWEA-2007-Final-Paper.pdf [^5]: https://www.euramet.org/download?tx_eurametfiles_download%5Baction%5D=download\&tx_eurametfiles_download%5Bcontroller%5D=File\&tx_eurametfiles_download%5Bfiles%5D=42776\&tx_eurametfiles_download%5Bidentifier%5D=%252Fdocs%252FPublications%252Fcalguides%252FI-CAL-GUI-025_Calibration_Guide_No._25_web.pdf\&cHash=d659a55ad3ec945d6eed296e75eacbbd [^6]: https://www.dantecdynamics.com/wp-content/uploads/2020/08/pratical-guide-how-to-measure-turbulence.pdf [^7]: https://www.mouser.com/catalog/additional/ATS_Qpedia_Dec07_Understanding%20hot%20wire%20amemometry9.pdf [^8]: https://people.eng.unimelb.edu.au/imarusic/publications/Edited Papers 2014/Talluru_MST_2014.pdf [^9]: https://papers.ssrn.com/sol3/Delivery.cfm/19206efa-3e84-4ba0-ab93-969505ef812b-MECA.pdf?abstractid=4821368\&mirid=1 [^10]: https://www.surreysensors.com/wp-content/uploads/2025/02/AN-0128-2025R1.pdf [^11]: https://www.sciencedirect.com/science/article/abs/pii/S0894177707001112 [^12]: https://web.iitd.ac.in/~pmvs/courses/mel705/hotwire1.pdf [^13]: https://pubs.aip.org/aip/pof/article/37/5/057104/3346715/A-modified-heat-transfer-law-for-hot-wire [^14]: https://nvlpubs.nist.gov/nistpubs/jres/15/jresv15n6p575_A1b.pdf [^15]: https://people.eng.unimelb.edu.au/imarusic/publications/Edited Papers 2022/Investigation of cold wire spatial_Xia.%20Y._Int.%20J.%20Heat%20and%20Fluid%20Flow.pdf [^16]: https://www.sciencedirect.com/science/article/pii/S026322412401114X [^17]: https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/hotwire-spatial-resolution-issues-in-wallbounded-turbulence/2CC580DB75447C0ADC710596DA2C131D [^18]: https://arc.aiaa.org/doi/10.2514/1.J055801 [^19]: https://www.fire.tc.faa.gov/pdf/fsr-0167.pdf [^20]: https://web.iitd.ac.in/~pmvs/courses/mel705/hotwire2.pdf [^21]: http://icing.ae.illinois.edu/papers/99/AIAA-0021-8669.pdf [^22]: https://www.tesscorn-aerofluid.com/wp-content/uploads/2015/09/Tao-Systems-Hot-Wire-Constant-Voltage-Anemometer-1.pdf [^23]: https://pmc.ncbi.nlm.nih.gov/articles/PMC7303135/ [^24]: https://asmedigitalcollection.asme.org/gasturbinespower/article/142/3/031007/975414/A-Probabilistic-Uncertainty-Estimation-Method-for [^25]: https://pmc.ncbi.nlm.nih.gov/articles/PMC10747627/ [^26]: https://www.bipm.org/documents/20126/2071204/JCGM_100_2008_E.pdf [^27]: https://www.ukas.com/wp-content/uploads/2023/05/M3003-The-expression-of-uncertainty-and-confidence-in-measurement.pdf [^28]: https://arxiv.org/abs/2401.09492 [^29]: https://www.sciencedirect.com/science/article/pii/S0894177721000546 [^30]: https://www.politesi.polimi.it/retrieve/a81cb05c-16cf-616b-e053-1605fe0a889a/Tesi_Roberto_Cosentino.pdf [^31]: http://ui.adsabs.harvard.edu/abs/2020MeScT..31d5302H/abstract [^32]: https://www.sciencedirect.com/science/article/abs/pii/S0017931098003718 [^33]: https://oa.upm.es/88316/1/5474731.pdf [^34]: https://pubs.aip.org/aip/pof/article/36/10/105192/3318280/A-simple-hot-wire-temperature-drift-correction [^35]: https://orbit.dtu.dk/files/7712116/DTU_Wind_Energy_E_0006_EN_.pdf [^36]: https://apps.dtic.mil/sti/tr/pdf/ADA078284.pdf [^37]: https://www.sciencedirect.com/topics/engineering/hot-wire-anemometer [^38]: https://www.sciencedirect.com/science/article/abs/pii/S0142727X22000017 [^39]: https://journal.gpps.global/Comparison-of-LDA-and-hot-wire-measurements-at-moderate-subsonic-mach-numbers,167521,0,2.html [^40]: https://www.sciencedirect.com/science/article/abs/pii/S0955598607000192 [^41]: https://lsi-lastem.com/blog/lsi-lastems-iso-17025-accredited-lab/ [^42]: https://www.scribd.com/document/730215922/Uncertainty-in-Anemometer-Calibration-geometry-rev2-1 [^43]: https://atslab.com/calibration/environmental-equipment/anemometer-calibration/ [^44]: https://www.sciencedirect.com/science/article/pii/S1290072924007890 [^45]: https://www.sciencedirect.com/science/article/abs/pii/S0263224122012350 [^46]: https://fluidhandlingpro.com/fluid-process-technology/fluid-flow-control-measurement/which-anemometer-should-i-use/ [^47]: https://labcal.co.uk/air-velocity-calibration/ [^48]: https://www.nist.gov/document/2008mscairpdf [^49]: https://www.bipm.org/documents/20126/2071204/JCGM_GUM_6_2020.pdf [^50]: https://www.calibrate.co.uk/calibration/air-velocity/about-anemometers/anemometers-calibration/ [^51]: https://www.thermopedia.com/de/content/853/ [^52]: https://www.extrica.com/article/23637 [^53]: https://scholarsmine.mst.edu/cgi/viewcontent.cgi?article=1071\&context=sotil [^54]: https://www.instrumentchoice.com.au/blogs/news/what-is-a-hot-wire-anemometer [^55]: https://amslaurea.unibo.it/id/eprint/17957/1/Master_Thesis_Lorenzo_Lazzarini.pdf [^56]: https://www.spiedigitallibrary.org/conference-proceedings-of-spie/3706/1/Hot-wire-calibration-in-a-nonisothermal-incompressible-pressure-variant-flow/10.1117/12.356955.full [^57]: http://lmfa.ec-lyon.fr/IMG/pdf/fontanetolmfa2017.pdf?1234%2Fa60e31c16bb867455f1eb980090cc0553ed25f99 [^58]: https://www.advancedtools.com.au/blogs/blog/what-is-a-hotwire-anemometer-essential-guide-to-airflow-measurement [^59]: https://www.sciencedirect.com/science/article/pii/S2212540X20300596 [^60]: https://ppl-ai-code-interpreter-files.s3.amazonaws.com/web/direct-files/34e4e69d827ae62ec0f80bbb913a22f9/3e28c955-4f6b-4239-9c6a-9fd561e60ec6/d1723220.csv """) return @app.cell(hide_code=True) def _(mo): mo.md(r""" # Uncertainty Budget of Hot-Wire Anemometer Isothermal turbulent air flow in a wind tunnel with temperature rise due to friction. Evaluation according to the 8-step GUM method. """) return @app.cell def _(): import numpy as np import pandas as pd # 1. Define standard uncertainties (%) for each source u = { # Calibration 'Reference velocity': 0.35, 'Reference temperature': 0.75, 'Calibration curve fitting': 1.25, 'Calibration drift': 3.00, 'Temperature compensation': 2.00, # Instrumentation 'Wire sensor characteristics': 1.25, 'Wire positioning': 1.00, 'CTA electronics': 0.65, 'A/D conversion': 0.06, 'Signal conditioning': 0.30, # Environmental 'Temperature drift': 3.50, 'Density variations': 1.25, 'Humidity effects': 1.25, 'Pressure fluctuations': 0.30, 'Flow non-uniformity': 0.60, 'Temperature rise friction': 3.00, # Resolution 'Wire length effects': 3.00, 'Frequency response': 6.00, 'Wire thermal inertia': 1.25, 'Probe interference': 1.25, # Processing 'Data acquisition noise': 0.30, 'Statistical sampling': 1.25, 'Turbulence corrections': 1.25, 'Non-isothermal corrections': 6.00 } # 2. Compute combined standard uncertainty u_c u_c = np.sqrt(sum(val**2 for val in u.values())) # 3. Compute expanded uncertainty U k = 2 U = k * u_c # 4. Identify dominant sources dominant = sorted(u.items(), key=lambda x: x[1], reverse=True)[:5] # 5. Summarize results results = { 'Combined standard uncertainty (%)': round(u_c, 2), 'Expanded uncertainty (%)': round(U, 2), 'Top 5 contributors': dominant } # pd.DataFrame(results) return @app.cell(hide_code=True) def _(mo): mo.md(r""" ### Results - **Combined standard uncertainty**: \(u_c \approx 11.42\%\) - **Expanded uncertainty (k=2)**: \(U \approx 22.85\%\) **Top 5 uncertainty sources**: 1. Frequency response (6.00%) 2. Non-isothermal corrections (6.00%) 3. Temperature drift (3.50%) 4. Calibration drift (3.00%) 5. Temperature rise due to friction (3.00%) """) return if __name__ == "__main__": app.run()