A new method for measuring the heating rate (defined as the time rate of change of temperature) and estimating heat flux from the heating rate is proposed. The example problem involves analytic heat conduction in a one-dimensional slab, where the measurement location of temperature or heating rate coincides with the location of the estimated heat flux. The new method involves the solution to a Volterra equation of the second kind, which is inherently more stable than Volterra equations of the first kind. The solution for heat flux from a measured temperature is generally a first kind Volterra equation. Estimates from the new approach are compared to estimates from measured temperatures. The heating rate measurements are accomplished by leveraging the temperature dependent decay rate of thermographic phosphors (TGP). Results indicate that the new data-reduction method is far more stable than the usual minimization of temperature residuals, which results in errors that are 1.5–12 times larger than those of the new approach. Furthermore, noise in TGP measurements was found to give an uncertainty of 4% in the heating rate measurement, which is comparable to the noise introduced in the test case data. Results of the simulations and the level of noise in TGP measurements suggest that this novel approach to heat flux determination is viable.
Issue Section:
Conduction
1.
Holmberg
, D. G.
, and Diller
, T. E.
, 1995, “High-frequency heat flux sensor calibration and modeling
,” J. Fluids Eng.
0098-2202, 117
, pp. 659
–664
.2.
Piccini
, E.
, Guo
, S. M.
, and Jones
, T. V.
, 2000, “The development of a new direct-heat-flux gauge for heat-transfer facilities
,” Meas. Sci. Technol.
0957-0233, 11
, pp. 342
–349
.3.
Diller
, T. E.
, and Kidd
, C. T.
, 1997, “Evaluation of numerical methods for determining heat flux with a null point calorimeter
,” in Proceedings of the 42nd International Instrumentation Symposium
, Research Triangle Park, NC, pp. 251
–262
.4.
Buttsworth
, D. R.
, and Jones
, T. V.
, 1997, “Radial conduction effects in transient heat transfer experiments
,” Aeronaut. J.
0001-9240, 101
, pp. 209
–212
.5.
Ireland
, P. T.
, and Jones
, T. V.
, 2000, “Liquid crystal measurements of heat transfer and surface shear stress
,” Meas. Sci. Technol.
0957-0233, 11
, pp. 969
–986
.6.
Newton
, P. J.
, Yan
, Y.
, Stevens
, N. E.
, Evatt
, S. T.
, Lock
, G. D.
, and Owen
, J. M.
, 2003, “Transient heat transfer measurements using thermochromic liquid crystal. Part 1: An improved technique
,” Int. J. Heat Fluid Flow
0142-727X, 24
, pp. 14
–22
.7.
Beck
, J. V.
, Blackwell
, B.
, and St. Claire
, Jr., C. R.
, 1985, Inverse Heat Conduction: Ill-Posed Problems
, Wiley–Interscience
, NY.8.
Guo
, S. M.
, Lai
, C. C.
, Jones
, T. V.
, Oldfield
, M. L. G.
, Lock
, G. D.
, and Rawlinson
, A. J.
, 1998, “The application of thin-film technology to measure turbine-vane heat transfer and effectiveness in a film-cooled, engine-simulated environment
,” Int. J. Heat Fluid Flow
0142-727X, 19
, pp. 594
–600
.9.
Dinu
, C.
, Beasley
, D. E.
, and Figliola
, R. S.
, 1998, “Frequency response characteristics of an active heat flux gage
,” J. Heat Transfer
0022-1481, 120
, pp. 577
–582
.10.
Hadamard
, J.
, 1923, Lectures on Cauchys Problems in Linear Partial Differential Equations
, Yale University Press
, New Haven, CT.11.
Kirsch
, A.
, 1996, An Introduction to the Mathematical Theory of Inverse Problems
, Springer
, Berlin, Vol. 120
.12.
Cook
, W. J.
, 1970, “Determination of heat transfer rates from transient surface temperature measurements
,” AIAA J.
0001-1452, 8
, pp. 1366
–1368
.13.
Kendall
, D. N.
, and Dixon
, W. P.
, 1966, “Heat transfer measurements in a hot shot wind tunnel
,” IEEE Trans. Aerosp. Electron. Syst.
0018-9251, AES-3
, pp. 596
–603
.14.
Walker
, D. G.
, and Scott
, E. P.
, 1997, “Evaluation of estimation methods for high unsteady heat fluxes from surface measurements
,” AIAA J.
0001-1452, 12
, pp. 543
–551
.15.
Buttsworth
, D. R.
, and Jones
, T. V.
, 1998, “A fast-response high spatial resolution total temperature probe using a pulsed heating technique
,” J. Turbomach.
0889-504X, 120
, pp. 612
–617
.16.
Dunn
, M. G.
, George
, W. K.
, Rae
, W. J.
, Woodward
, S. H.
, Moller
, J. C.
, and Seymour
, P. J.
, 1986, “Heat flux measurements for the rotor of a full-stage turbine: Part II: Description of analysis technique and typical time-resolved measurements
,” ASME J. Turbomach.
0889-504X, 108
, pp. 98
–107
.17.
Ehrich
, F. F.
, 1954, “Differentiation of experimental data using least squares fitting
,” J. Aeronaut. Sci.
0095-9812, 22
, pp. 133
–134
.18.
Tikhonov
, A. N.
, and Arsenin
, V. Y.
, 1977, Solutions of Ill-Posed Problems
, V. H. Winston & Sons
, Washington, D.C.19.
George
, W. K.
, Rae
, W. J.
, Seymour
, P. J.
, and Sonnenmeier
, J. R.
, 1987, “An evaluation of analog and numerical techniques for unsteady and heat transfer measurements with thin film gauges in transient facilities
,” in Proceedings of the ASME/JSME Thermal Engineering Joint Conference
, Honolulu
, HI, Vol. 2
, pp. 611
–617
.20.
Walker
, D. G.
, Scott
, E. P.
, and Nowak
, R. J.
, 2000, “Estimation methods for 2D conduction effects of shock-shock heat fluxes from temperature measurements
,” AIAA J.
0001-1452, 14
, pp. 533
–539
.21.
Frankel
, J. I.
, and Keyhani
, M.
, 1999, “Inverse heat conduction: The need for heat rate data for design and diagnostic purposes
,” in Proceedings of the 18th LASTED International Conference on Modeling, Identification and Control
, Innsbruck
, Austria, February.22.
Frankel
, J.
, and Osborne
, G.
, 2004, “Motivation for the development of heating/cooling rate and heat flux rate sensors for engineering applications
,” in Proceedings of the 42nd AIAA Aerospace Sciences Meeting and Exhibit
.23.
Lamm
, P. K.
, 1995, “Future-sequential regularization methods for ill-posed volterra equations
,” J. Math. Anal. Appl.
0022-247X, 195
, pp. 469
–494
.24.
Walker
, D. G.
, and Schetz
, J. A.
, 2003, “A new technique for heat flux determination
,” in Proceedings of the ASME Summer Heat Transfer Conference
, Las Vegas
, NV.25.
Allison
, S. W.
, and Gillies
, G. T.
, 1997, “Remote thermometry with thermographic phosphors: Instrumentation and applications
,” Rev. Sci. Instrum.
0034-6748, 68
, pp. 2615
–2650
.26.
Sholes
, R. R.
, 1980, “Fluorescent decay thermometry with biological applications
,” Rev. Sci. Instrum.
0034-6748, 51
, pp. 882
–884
.27.
Feist
, J. P.
, and Heyes
, A. L.
, 2000, “The characterization of Y2O2S:Sm powder as a thermographic phosphor for high temperature applications
,” Meas. Sci. Technol.
0957-0233, 11
, pp. 942
–947
.28.
Allison
, S. W.
, Cates
, M. R.
, Noel
, B. W.
, and Gillies
, G. T.
, 1988, “Monitoring permanent-magnet motor heating with phosphor thermometry
,” IEEE Trans. Instrum. Meas.
0018-9456, 37
, pp. 637
–641
.29.
Necati Özişik
, M.
, 1968, Boundary Problems of Heat Conduction
, Dover
, New York.30.
Kress
, R.
, 1989, Linear Integral Equations
, Vol. 82
in Applied Mathematical Sciences
, Springer-Verlag
, Berlin.31.
Corduneanu
, C.
, 1991, Integral Equations and Applications
, Cambridge University Press
, Cambridge.32.
Stolz
, Jr. G.
, 1960, “Numerical solutions to an inverse problem of heat conduction for simple shapes
,” J. Heat Transfer
0022-1481, 82
, pp. 20
–26
.33.
Shionoya
, S.
, and Yen
, W. M.
, editors, 1999, Phosphor Handbook
, CRC Press
, Boca Raton.34.
Beck
, J. V.
, and Arnold
, K. J.
, 1977, Parameter Estimation in Engineering and Science
, Wiley
, New York.35.
Hogg
, Robert V.
, and Ledolter
, Johannes
, 1992, Applied Statistics for Engineers and Physical Scientists
, 2nd ed., Macmillan
, NY.36.
Belding
, D. F.
, and Mitchell
, K. J.
, 1991, Foundation of Analysis
, Prentice–Hall
, NJ.Copyright © 2005
by American Society of Mechanical Engineers
You do not currently have access to this content.
