Shock Waves

, Volume 23, Issue 2, pp 177–188 | Cite as

Theoretical development of a new surface heat flux calibration method for thin-film resistive temperature gauges and co-axial thermocouples

Original Article


This paper presents a theoretically developed and computationally demonstrated surface heat flux calibration method applicable to thin-film resistive temperature gauges and co-axial thermocouples. For this study, the physical situation of interest involves hypersonic shock-tunnel studies. For experiments instrumented with these gauges, constant thermophysical properties are assumed since small temperature variations normally occur in the short-duration run times. Extraction of the net surface heat flux is acquired by resolving a newly formulated first-kind Volterra integral equation that contains calibration data. The proposed calibration method is based on an inverse approach which contrasts system identification methods. Several key advantages to this approach are discussed and demonstrated in the context of these gauges. Advantages of the proposed approach include (a) only one unknown “regularization” parameter is required; (b) estimation of the optimal regularization parameter is systematically and theoretically developed and demonstrated through the energy residuals, (c) computational coding is minimal and computer run times are short, and (d) results indicate robustness, stability and accuracy in the methodology. This calibration formulation and its subsequent regularized numerical method do not explicitly require the thermal effusivity, \(\sqrt{\rho C k}\) owing to its input–output based derivation.


Heat flux Calibration Thin-film sensors Co-axial thermocouples Inverse analysis 


  1. 1.
    Sahoo, N.: Experiments on a blunt cone model in hypersonic shock tunnel. In: 16th Australasian Fluid Mechanics Conference, Gold Coast, Australia, 2–7 Dec 2007Google Scholar
  2. 2.
    Olivier, H.: Thin-Film Gauges and Coaxial Thermocouples for Measuring Transient Temperatures (Description-gauges2.pdf, from
  3. 3.
    George, W.K., Rae, W.J., Woodward, S.H.: An evaluation of analog and numerical techniques for unsteady heat transfer measurements with thin-film gauges in transient facilities. Exp. Thermal Fluid Sci. 4, 333–342 (1991)CrossRefGoogle Scholar
  4. 4.
    Buttsworth, D.R., Jones, T.V.: Transient temperature probe measurements in a Mach 4 nitrogen jet. Exp. Fluids 37, 137–145 (2004)CrossRefGoogle Scholar
  5. 5.
    Buttsworth, D.R.: Heat transfer during transient compression: measurements and simulations. Shock Waves 12, 87–91 (2002)CrossRefGoogle Scholar
  6. 6.
    Piccini, E., Guo, S.M., Jones, T.V.: The development of a new direct-heat-flux gauge for heat transfer facilities. Measur. Sci. Technol. 11, 342–349 (2000)CrossRefGoogle Scholar
  7. 7.
    Gulhan, A.: Heat flux measurements in high enthalpy flows. Defense Technical Information Center Compilation Part Notice ADP010750 (unclassified) (1999)Google Scholar
  8. 8.
    Buttsworth, D.R., Stevens, R., Stone, C.R.: Eroding Ribbon thermocouples: impulse response and transient heat flux analysis. Measur. Sci. Technol. 16, 1487–1494 (2005)CrossRefGoogle Scholar
  9. 9.
    Bouchard, P.O., Chambers, H.F.: Description of the 120 inch hypersonics shock tunnel and anticipated performance. Technical Report ARAPL-TR-65-129 (1966)Google Scholar
  10. 10.
    Lu, F.K., Wilson, D.R.: Detonation driver for enhancing shock tube performance. Shock Waves 12, 457–468 (2003)CrossRefGoogle Scholar
  11. 11.
    Frankel, J.I., Keyhani, M., Elkins, B.: Surface heat flux prediction through physics-based calibration. Part 1: theory. In: 50th AIAA Aerospace Sciences Meeting and Exhibit, Nashville, TN, 9–12 January 2012Google Scholar
  12. 12.
    Loehle, S., Battaglia, J.L., Batsale, J.C., Bourserau, F., Conte, D., Jullien, P., Ootegem, B.V., Couzi, J., Lasserre, J.P.: Estimation of high heat flux in supersonic plasma flows. IEEE 4, 5366–5373 (2006)Google Scholar
  13. 13.
    Loehle, S., Battaglia, J.L., Jullien, P., Ootegem, B.V., Couzi, J., Lasserre, J.P.: Improvement of high heat flux measurement using a null point calorimeter. AIAA J. Spacecraft Rocket 45(1), 76–81 (2008)CrossRefGoogle Scholar
  14. 14.
    Gardarein, J.L., Battaglia, J.L., Loehle, S.: Heat flux sensor calibration using noninteger system identification: theory, experiment and error analysis. Rev. Sci. Instrum. 80, 205102 (2009)CrossRefGoogle Scholar
  15. 15.
    Loehle, S., Frankel, J.I.: Physical insight into system identification parameters applied to inverse heat conduction problems. In: AIAA 50th AIAA Aerospace Sciences Meeting and Exhibit, Nashville, TN, 9–12 January 2012 (AIAA-2012-0809)Google Scholar
  16. 16.
    Cook, W.J., Felderman, E.J.: Reduction of data from thin-film heat transfer gages: a concise numerical technique. AIAA J. 4, 561–562 (1966)CrossRefGoogle Scholar
  17. 17.
    Lu, F.K., Kinnear, K.M.: Characterization of thin film heat flux gauges. J. Thermophys. Heat Transf. 13(4), 548–550 (1999)CrossRefGoogle Scholar
  18. 18.
    Diller, T.E.: Advances in heat flux measurements. In: Advances in Heat Transfer, vol. 23. Academic Press, New York, pp. 279–368 (1993)Google Scholar
  19. 19.
    Keltner, N.R.: Heat flux measurements: theory and applications. In: Thermal Measurements in Electronic Cooling, pp. 273–320. CRC, New York (1997)Google Scholar
  20. 20.
    Flaherty, W., Austin, J.M.: Comparative surface heat transfer measurements in hypervelocity flow. J. Thermophys. Heat Transf. 25(1), 180–183 (2011)CrossRefGoogle Scholar
  21. 21.
    Ozisik, M.N.: Heat Conduction. Wiley, New York (1980)Google Scholar
  22. 22.
    Grigull, U., Sandner, H.: Heat Conduction, p. 18. Hemisphere Publishing Co., New York (1984)Google Scholar
  23. 23.
    Beck, J.V., Blackwell, B., St. Clair, C.R. Jr.: Inverse Heat Conduction. Wiley, New York (1985)Google Scholar
  24. 24.
    Elkins, B.E., Keyhani, M., Frankel, J.I.: Surface heat flux prediction through physics-based calibration. Part 2: experimental verification. In: 50th AIAA Aerospace Sciences Meeting and Exhibit, Nashville, TN, 9–12 January 2012Google Scholar
  25. 25.
    Woodbury, K.A.: Effect of thermocouple sensor dynamics on surface heat flux predictions obtained via inverse heat transfer analysis. Int. J. Heat Mass Transf. 33(2), 2641–2649 (1990)Google Scholar
  26. 26.
    Frankel, J.I., Elkins, B., Keyhani, M.: Rate-based sensing concepts for heat flux and property estimation; and, transition detection. In: 16th AIAA/DLR/DGLR International Space Planes and Hypersonic Systems and Technologies Conference, Bremen, Germany. AIAA Paper 09-7303, 19–22 October 2009Google Scholar
  27. 27.
    Blanchat, T.K., Humphries, L.L., Gill, W.: Sandia heat flux gauge thermal response and uncertainty models. Sandia Report, SAND 2000-11 (2000) Google Scholar
  28. 28.
    Frankel, J.I.: Regularization of inverse heat conduction by combination of rate sensor analysis and analytic continuation. J. Eng. Math. 57, 181–198 (2007)Google Scholar
  29. 29.
    Frankel, J.I., Arimilli, R.V.: Inferring convective and radiative loads from transient surface temperature measurements in the half space. Inverse Problems Sci. Eng. 15(5), 463–488 (2007)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Linz, P.: Analytical and Numerical Methods for Volterra Equations. SIAM, Philadelphia (1985)MATHCrossRefGoogle Scholar
  31. 31.
    Lamm, P.K.: A survey of regularization methods for first-kind Volterra equations. In: Colton, D., Engl, H.W., Louis, A., McLaughlin, J.R., Rundell, W. (eds.) Surveys on Solution Methods for Inverse Problems, pp. 53–82. Springer, New York (2000)CrossRefGoogle Scholar
  32. 32.
    Frankel, J.I., Keyhani, M., Taira, K.: In-phase error estimation of experimental data and optimal first derivatives. AIAA J. 42(5), 1017–1024 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Mechanical, Aerospace and Biomedical Engineering DepartmentUniversity of TennesseeKnoxvilleUSA

Personalised recommendations