Dr. Jay I. Frankel

Research Interests:

Heat Transfer in Hypersonics, Slug Calorimentry and Thin-Film Gauges, Inverse Heat Transfer Problems, Thermal Measurements, Calibration Mathematics, Uncertainty Analysis, Sensitivity Analysis, Parameter Estimation, Applied Mathematics, Vibrations

Memberships and Services:

• AIAA Thermophysics Technical Committee (2018-2021)
• AIAA Special Awards, AIAA TN (May 2018): For “recognizing your work in engineering analysis and the development of CIEM”
• AIAA Technical Chair Thermophysics Conference, AIAA Aerospace Science Meeting (ASM)- Nashville, TN January 2012.
• AIAA Thermophysics Technical Committee (2009-2012)
• AIAA TC-Thermophysics Liaison to TC-Ground Testing (2009-2012)
• Guest Professor, Ocean University of China, Qingdao, China (2009-2012)
• AIAA General H.H. Arnold Award, AIAA TN (2010): General H. H. Arnold Award is granted for outstanding contributions in the advancement of thermal analysis in harsh rocket environments and high speed propulsion systems.
• Distinguished Visiting Scientist and Faculty, Oak Ridge National Laboratory (ORNL) (July 07–Jan 08)
• AIAA Associate Fellow, American Institute of Aeronautics and Astronautics (2005)
• AIAA Thermophysics Technical Committee (2003-2006)
• AIAA Thermophysics Technical Committee (2000-2003)
• Fellow, Computational Mechanics, Wessex Institute of Technology, Southampton, UK (2000)

• Frankel, J.I., Keyhani, M., Arimilli, R.V., and J. Wu “Rate-based sensors for advance real-time analysis and diagnostics”, U.S. Patent No. 7539587, May 26, 2009.

• “New Paradigms for Inverse Heat Conduction Problems: Creative analytics and experiments utilizing advanced technologies“, PI/PD: J.I. Frankel, National Science Foundation, CBET-1703442, Amount Received: $302,314, Duration: 1 Aug. 2017- 31 July 2021.
• “A High Temperature, High Heat Flux Electrical Heater Test Facility for Rapid Evaluation of Surface Thermal Conditions, Alternative Sensing and Cross Calibration”, PI: J.I. Frankel, subcontracted from University of Dayton Research Institute, Amount Received: $208,784, Duration: 1 January 2018- 31 December 2020.
• “Total Heat Transfer Calibration in Multilayer Plates”, PI/PD: J.I. Frankel, Co-Pi: M. Keyhani, MBDA-France, Amount Received: $208,259 (186,780 Euro), Award Period: Sept. 1, 2015- March 1, 2019.
• “A Novel Calibration Method for Estimating Transient Surface Heat Flux under Hypersonic Heating Conditions for Global Reach Vehicles”, PI/PD J.I. Frankel, and Co-Pi: M. Keyhani, Air Force Research Laboratory (WP), Contract Number: FA8650-13C-2326, Amount Received: $374,181, Award Period: Dec. 1, 2012-Dec 10, 2017. Plus supplement of 14K for UG support.
• “Transformative Calibration Method for Prediction of Surface Heat Flux”, PI/PD J.I. Frankel, Co-Pi: M. Keyhani, National Science Foundation (CBET-1234419), Amount Received: $249,915, Award Period: Aug. 15, 2012-August 14, 2016.
• “EAGER: Experimental Verification to a Transformative Calibration Method”, PI/PD J.I. Frankel, Co-Pi: M. Keyhani, National Science Foundation (CBET-1137625 Combustion, Fire and Plasma Systems), Amount Received: $57,503: Breakdown: $48,500 (CBET), with supplement $9,003 (OISE-1148763); Award Period: Aug. 1, 2011-July 31, 2012.
• “EAGER: Application of Calibration Convolution Integrals to Diffusion Transport”, PI/PD J.I. Frankel, Co-Pi: M. Keyhani, National Science Foundation (CBET-1153476), Amount Requested: $28,954, Award Period: September 1, 2011-August 31, 2012.
• “Heating rate sensor and analytic tools for the prediction of surface heat flux and temperature of TPS via in-depth sensor data”, PI/PD J.I. Frankel, Co-Pi: M. Keyhani, NASA (Contract Number: NNX10AN35A), Amount Budgeted: $1,059,300.00 (Amounts Awarded: NASA: $522,286, UTK cost share: $537,014), Award Period: August 1, 2010-July 31, 2014.
• “Time Derivative of Temperature Sensor Development for Advanced Thermal Science Applications” Aerospace Testing Alliance, Contract Number: 09-16-FRANKEL, Amount Awarded: $45,000, Award Period: 2/1/09-12/31/09.
• “Rate-Based Sensor Development for Advancing Heat Transfer Measurements” (PI/PD J.I. Frankel, Co-PI’s: M. Keyhani, R. Arimilli, and J. Wu), National Science Foundation, CBET-0601236, Amount: $114,855, Award Period: April 15, 2006-September 15, 2007.
• “Inverse Process Analysis for the Acquisition of Thermophysical Property Data” (PI/PD J.I. Frankel (UTK), Co-PI’s: A. Sabau (ORNL), W. Porter (ORNL), Department of Energy (DE-FC07-01ID14249), Amount Awarded: $750,000 (UTK: $345,000, ORNL, $405,000), Award Period: October 1, 2001-September 30, 2004.
• “A New Unified Space/Time Treatment for Solving Direct and Thermal Design Problems in Radiative and Conductive Transport”, National Science Foundation (CTS-96-19192) and Sandia National Laboratories, Amount Awarded: $170,150 (NSF-CTS: $95,150 SNL: $75,000), Award Period: Nov. 1, 1997-October 31, 2001 (includes NSF-REU supplement of $5,000).

1. Frankel, J.I., Geraets, R.T.P. (Oxford), McGilvray, M. (Oxford) and H.C. Chen (Rheem), “Alternative Data Reduction Equation for Shock Tunnel Experiments Using Diamond Slug Calorimetry”, AIAA J. Thermophysics and Heat Transfer (under “minor” revision).
2. Frankel, J.I., and K. Ekici, “Revisiting Slug Calorimeters for Hypersonics”, Aerospace Science and Technology Journal, (accepted, 3/2020).
3. Ekici, K., Djeddi, R., Li, H., Hang, and J.I. Frankel, “Modeling Periodic and Non-periodic Response of Dynamical Systems Using an Efficient Chebyshev-Based Time-Spectral Approach” J. Computational Physics, (accepted 5/2020).
4. Frankel, J.I. and H.C. Chen, “Analytical Developments and Experimental Validation of a Thermocouple Model through an Experimentally Acquired Impulse Response Function”, Int. J. Heat and Mass Transfer, Vol. 141 (2019), pp. 1301-1314. doi: 10.1016/j.ijheatmasstransfer.2019.05.098
5. Frankel, J.I., and H.C. Chen, “Heat Flux Data Reduction using a Pre-conditioner Test Function for Thin Film Analysis”, AIAA J. Thermophysics and Heat Transfer (in press). http://arc.aiaa.org/doi/abs/10.2514/1.T5744
6. Chen, H.C., and J.I. Frankel, “Estimating the Thermal Effusivity for Coaxial Thermocouple Using a Quantified Aluminum Nitride Heater”, AIAA J. Thermophysics and Heat Transfer (in press) https://doi.org/10.2514/1.T5743
7. Chen, H.C., Frankel, J.I. and M. Keyhani, “Two-Probe Calibration Integral Equation Method for Nonlinear Inverse Heat Conduction Problem of Surface Temperature Estimation”, Numerical Heat Transfer, Part B, (minor revision due Jan 2020).
8. Frankel, J.I., Chen, H.C., and K. Mathew, “Design-to-Validation of a New Heat Flux Source with Uncertainty Analysis”, AIAA J. Thermophysics and Heat Transfer, Vol. 33, No. 2 (2019), pp. 517-535.http://arc.aiaa.org/doi/abs/10.2514/1.T5519.
9. Myrick, J.A., Keyhani, M, and J.I. Frankel, “Calibration of a Plate Sensor for Determination of Total Heat Transfer into a Surface with a Spatially Varying Heat Flux”, Experimental Thermal and Fluid Science Vol. 97 (2018), pp 145-159. doi: 10.1016/j.expthermflusci.2018.04.005
10. Chen, H.C., Frankel, J.I. and M. Keyhani, “Two-Probe Calibration Integral Equation Method for Nonlinear Inverse Heat Conduction Problem of Surface Heat Flux Estimation”, Int. J. Heat Mass Transfer, Vol. 121 (2018), pp. 246-26, doi: 10.1016/j.ijheatmasstransfer.2017.12.072
11. Chen, H.C., Frankel, J.I. and M. Keyhani, “Nonlinear Inverse Heat Conduction Problem of Surface Temperature Estimation by the Calibration Integral Equation Method”, Numerical Heat Transfer, Part B Vol. 73, Issue 5 (2018) pp. 263-291 doi.org/10.1080/10407790.2018.1464316
12. Chen, H.C., Frankel, J.I. and M. Keyhani, “Nonlinear Inverse Heat Conduction: Digitally Filtered Space Marching with Phase-Plane and Cross-Correlation Analysis”, Numerical Heat Transfer, Part B, Vol. 72, Issue 2, (2017), pp. 109-129. doi: 10.1080/10407790.2017.1347004.
13. Chen, Y-Y, Frankel, J.I. and M. Keyhani, “A New Front Surface Heat Flux Calibration Method for 1-D Nonlinear Thermal System with a Time-Varying Back Boundary Condition”, J. Engineering Mathematics, Vol. 105, Issue 1 (2017) pp. 157-187. doi: 10.1007/s10665-016-9888-0.
14. Myrick, J.A., Keyhani, M, and J.I. Frankel, “Determination of Surface Heat Flux and Temperature Using In-Depth Temperature Data – Experimental Verification”, Int. J. Heat and Mass Transfer, Vol. 111, (2017), pp. 982-998, doi.org/10.1016/j.ijheatmasstransfer.2017.04.029.
15. Frankel, J.I. and M. Keyhani, “Response Function Formulation for Calibration Integral Equation Method: Concept”, Journal of Engineering Mathematics, Vol. 110, Issue 1 (2018), pp. 75-95. DOI: 10.1007/s10665-017-9932-8.
16. Frankel, J.I., Chen, H.C., and M. Keyhani, “New Step Response Formulation for Inverse Heat Conduction”, AIAA J.Thermophysics and Heat Transfer, Vol. 31, No. 4 (2017), pp. 989-996 http://arc.aiaa.org/doi/abs/10.2514/1.T5067.
17. Frankel, J.I., and M. Keyhani, “Cross Correlation and Inverse Heat Conduction by a Calibration Method”, AIAA J.Thermophysics and Heat Transfer, Vol. 31, No. 3 (2017), pp. 746-756 http://arc.aiaa.org/doi/abs/10.2514/1.T5059.
18. Myrick, J.A., Keyhani, M, and J.I. Frankel, “Inverse Prediction of Temperature through Time Rescaling of High Temperature Experimental Data”, Journal of Thermal Science and Engineering Applications, Vol.8, No. 4 (2016) 041005 (12 pages) doi: 10.1115/1.4034093.
19. Chen, Y-Y., Keyhani, M. and J.I. Frankel, “A New Thermophysical Property Estimation Approach based on Calibration Equations and Rescaling Principle”, Journal of Thermal Science and Engineering Applications, Vol. 8, No. 2 (2015) pp. 021013-021023, doi: 10.1115/1.4032179.
20. Frankel, J.I. and M. Keyhani, “Inverse Ablation Analysis and the Calibration Integral Equation Method”, AIAA J.Thermophysics and Heat Transfer, Vol. 31, No. 1 (2017), pp. 10-19. doi: 10.2514/1.T4625.
21. Chen, Y-Y., Frankel, J.I., and M. Keyhani, “A Nonlinear, Rescaling Based Inverse Heat Conduction Calibration Method and Optimal Regularization Parameter Strategy”, AIAA J. Thermophysics and Heat Transfer, Vol. 30, No. 1 (2016), pp.67-88 (doi: 10.2514/1.T4572).
22. Chen, H.C., J.I. Frankel, and M. Keyhani, “Two-Dimensional Formulation for Inverse Heat Conduction Problems by the Calibration Integral Equation Method”, Applied Mathematical Modeling, Volume 40, Issues 13–14, (2016), pp. 6588–6603 (doi:10.1016/j.apm.2016.02.003).
23. Frankel, J.I. and D. Bottländer, “Acoustic Interferometry and the Calibration Integral Equation Method for Inverse Heat Conduction”, AIAA J. Thermophysics and Heat Transfer, Vol. 29, No. 4 (2015), pp. 862-869
24. Löhle, S. (Univ. Stuttgart) and J.I. Frankel, “Physical Insight into System Identification Parameters Applied to Inverse Heat Conduction Problems”, AIAA J. Thermophysics and Heat Transfer, Vol. 29, No. 3 (2015), pp 467-472, doi: 10.2514/1.T4431.
25. Frankel, J.I. and M. Keyhani, “Calibration Integral Equation Method for Two-Probe Inverse Heat Conduction” AIAA J. Thermophysics and Heat Transfer, Vol. 28, No. 3 (2014) pp. 548-553. doi: 10.2514/1.T4374.
26. Frankel, J.I., M. Keyhani, “Phase-Plane Analysis and Cross-Correlation for Estimating Optimal Regularization Parameters in Inverse Heat Conduction Problems”, AIAA J. Thermophysics and Heat Transfer, Vol. 28, No. 3 (2014) pp. 542-548. (doi: 10.2514/1.T4357).
27. Chen, Y.Y., Frankel, J.I., and M. Keyhani, “A New Nonlinear Surface Heat Flux Calibration Method based on Kirchhoff Transformation and Rescaling Principles”, Inverse Problems in Sciences and Engineering, Vol. 22, #8 (2014), pp. 1394-1421. (http://dx.doi.org/10.1080/17415977.2014.880906).
28. Frankel, J.I., and M. Keyhani, “Theoretical Development of a New Surface Heat Flux Calibration Method for Thin-Film Resistive Temperature Gauges and Co-Axial Thermocouples”, Shock Waves, Vol. 23, Issue 2, pp 177-188 (2013). doi: 10.1007/s00193-012-0418-3
29. Frankel, J.I. and M. Keyhani, “Nonlinear Inverse Calibration Heat Conduction through Property Physics”, AIAA J. Thermophysics and Heat Transfer, Vol. 28, No. 2, pp. 203-217 (2014)
30. Elkins, B., M. Keyhani, and J.I. Frankel, “Surface Heat Flux Prediction Through Physics-Based Calibration- Part 2: Experimental Verification, AIAA Journal of Thermophysics and Heat Transfer, Vol. 27, No. 2 , pp. 206-216 (2013).doi: 10.2514/1.T3918
31. Frankel, J.I., M. Keyhani, and B. Elkins, “Surface Heat Flux Prediction through Physics-Based Calibration- Part 1: Theory, AIAA Journal of Thermophysics and Heat Transfer, Vol. 27, No. 2, pp. 189-205 (2013). doi: 10.2514/1.T3917
32. Frankel, J.I., Keyhani, M, “Moving Low-Pass Gauss Filter with Automatically Defined Influence Region for Diffusive Studies”, AIAA J. Thermophysics and Heat Transfer, Vol. 26, #1, pp. 176-181 (2012).
33. Elkins, B.S., Keyhani, M., and J.I. Frankel ,”Global Time Method for the Inverse Heat Conduction Problem”, Inverse Problems in Science and Engineering, Vol. 20, Issue 5, pp. 651-664 (2012).
34. Frankel, J.I., M. Keyhani, R.V. Arimilli, and B. Elkins, “A New Rate-Based Sensor Approach for Measuring Thermal Diffusivity”, AIAA J. Thermophysics and Heat Transfer, Vol. 24, #4, pp. 811-817 (2010).
35. Frankel, J.I., M. Keyhani, R.V. Arimilli, and B. Elkins, “A New, Anisotropic, Two-Dimensional, Transient Heat Flux-Temperature Integral Relationship for Half-Space Diffusion”, Zeitschrift für angewandte Mathematik und Mechanik (ZAMM) , Vol.90, No. 2, pp. 161-170 (2010).
36. Elkins, B.E., M. G. Huang, and J.I. Frankel, “In-Situ Higher-Time Derivative of Temperature Sensors for Aerospace Heat Transfer”, Int. Journal of Thermal Sciences, Vol. 52, pp. 31-39 (2012).
37. Frankel, J.I., M. Keyhani, and R.V. Arimilli, “A New, Orthotropic, Two-Dimensional, Transient Heat Flux-Temperature Integral Relationship for Half-Space Diffusion”, AIAA J. Thermophysics and Heat Transfer , Vol. 24, #1, pp. 212-215 (2010).
38. Frankel, J.I., M. Keyhani, and M.G. Huang, “Local Sensitivity Analysis for the Heat Flux Temperature Integral Relationship in the Half Space”, Applied Mathematics and Computations, Vol. 217, pp. 363-375 (2010).
39. Frankel, J.I., M. Keyhani, and R. Arimilli, “New Three-Dimensional Heat Flux-Temperature Integral Relation for Half-Space Transient Diffusion”, Journal of the Franklin Institute, Vol. 347, pp. 1681-1688 (2010).
40. Kruttiventi, J, J. Wu and J.I. Frankel, “Real-Time Low-Noise Differentiation of Low Frequency Signals in Heat-Transfer Applications”, IEEE Transactions on Instrumentation and Measurements, Vol. 59, #3, pp. 596-603 (2010).
41. Frankel, J.I., “A New Asymptotic Temperature-Heat Flux Integral Relationship in Cylindric Coordinates”, AIAA J. Thermophysics and Heat Transfer, Vol. 23, #1, pp. 191-196 (2009).
42. Frankel, J.I., “Temperature-Heat Flux Integral Relationship in the Half-Space by Fourier Transforms”, AIAA J. Thermophysics and Heat Transfer, Vol. 22, #4, pp. 786-791 (2008).
43. Frankel, J.I., M. Keyhani, R.V. Arimilli, and J. Wu, “Locating Sudden Changes in Heat Flux using Higher-Temporal Derivative of Temperature”, AIAA. J. Spacecraft and Rockets, Vol. 45, #3, pp. 631-635 (2008).
44. Frankel, J.I., and C. Chisholm, “Integral Equation and Sensitivity Analyses of Creep Behavior for PVDF Thin Films”, Mechanics of Materials, Vol. 40, #7, pp 594-601 (2008).
45. Frankel, J.I and V.V. Kulish, “A New Flux Integral Relationship for Half-Space Transient Heat Conduction with Arbitrary Initial Condition”, Engineering Analysis with Boundary Elements, Vol. 32, pp. 162-167 (2008).
46. Frankel, J.I., R.V.Arimilli, M.Keyhani, and J. Wu, “Heating Rate dT/dt Measurements Developed from In-Situ Thermocouples using a Voltage-Rate Interface”, Int. Comm. of Heat and Mass Transfer , Vol. 35, pp. 885-891 (2008)
47. Frankel, J.I., M. Keyhani, R.V. Arimilli, and J. Wu, “A New Multidimensional Integral Relationship between Heat Flux and Temperature for Direct Internal Assessment of Heat Flux”, Zeitschrift für angewandte Mathematik und Physik (ZAMP), Vol. 59, #5, pp. 869-888 (2008).
48. Frankel, J.I., “Generalizing the Method of Kulish to One-Dimensional Unsteady Heat Conducting Slabs”, AIAA J. Thermophysics and Heat Transfer, vol. 20, #4, pp. 945-950 (2006).
49. Frankel, J.I., “Regularization of Inverse Heat Conduction by Combination of Rate Sensors Analysis and Analytic Continuation”, J. Engineering Mathematics, vol. 57, pp. 181-198 (2007).
50. Frankel, J.I., and R.V. Arimilli, “Inferring Convective and Radiative Heating Loads from Transient Surface Temperature Measurements in the Half-Space”, Inverse Problems in Sciences and Engineering vol. 15, No. 5, pp. 463-488 (2007).
51. Frankel, J.I., W.D. Porter, and A.S. Sabau, “Analysis of a Modified Push-Rod Dilatometer”, J. Thermal Analysis and Calorimetry, vol. 82, pp. 171-177, (2005).
52. Frankel, J.I., “Motivation for the Development of New Thermal Rate Sensors for Material Science Applications”, International Journal of Materials and Product Technology (Special Issue: Quenching and Distortion), Invited Paper, vol. 24, No. 1-4, pp. 199-206, (2005).
53. Frankel, J.I., and J.J. Lawless, “Numerically Stabilizing Ill-Posed Moving Surface Problems through Heat Rate Sensors”, AIAA J. Thermophysics and Heat Transfer, vol. 19, #4, pp. 587-592, (2005).
54. Frankel, J.I., G.E. Osborne, and K. Taira, Stabilization of Ill-Posed Problems through Thermal Rate Sensors”, AIAA J. Thermophysics and Heat Transfer, vol. 20, #2, pp. 238-246, (2006).
55. Frankel, J.I., and K. Taira, “Integral Equation Formulation and Error Estimation for Radial Flow Between Two Flat Disks”, Journal of Computational and Applied Mathematics, vol. 181, pp. 103-124, (2005).
56. Frankel, J.I., M. Keyhani, and K. Taira, “In-Phase Error Estimation of Experimental Data and Optimal First Derivatives”, AIAA Journal, vol. 42, #5, pp. 1017-1024, (2004).
57. Frankel, J.I., “Inverse Heat Conduction and Data-Type Issues”, Boundary Element Communications, vol. 11, #4, pp. 37-42, (2000).
58. Hale, S.W., M. Keyhani, and J.I. Frankel, “Design and Control of Interfacial Temperature Gradients in Solidification”, International Journal of Heat and Mass Transfer, vol. 43, #20, pp. 3795-3810, (2000).
59. Frankel, J.I., and M. Keyhani, “A Global Time Treatment for Inverse Heat Conduction Problems”, Journal of Heat Transfer, vol. 119, pp. 673-683 (1997).
60. Frankel, J.I., and G.E. Osborne, “A New Time Treatment for Solving Partial Integro-Differential Equations of Radiative Transport”, IMA Journal of Numerical Analysis, vol. 19, pp. 91-103 (1999).
61. Frankel, J.I. and M. Keyhani, “A New Approach for Solving Inverse Solidification Design Problems”, Numerical Heat Transfer, Part B, vol. 30, #2, pp. 161-178 (1996).
62. Frankel, J.I., “A Weighted-Residual Solution to an Inverse Stefan Design Problem with Imposed Physical Constraints”, Zeitschrift für angewandte Mathematik und Physik (ZAMP), vol. 47, pp. 456-466 (1996).
63. Frankel, J.I., “Direct Least-Squares Solutions to Integral Equations Containing Discrete Data”, AIAA Journal of Thermophysics and Heat Transfer, vol. 10, #1, pp. 182-186 (1996).
64. Frankel, J.I., “Implicit Solution Method for a Cumulative Variable Formulation to Radiative/Conductive Transport”, AIAA Journal of Thermophysics and Heat Transfer, vol. 9, #3, pp. 558-560 (1996).
65. Frankel, J.I., “A Residual-Minimization Least-Squares Method for Inverse Heat Conduction”, Computers and Mathematics with Applications, vol. 32, #4, pp. 117-130 (1996).
66. Frankel, J.I., “Cumulative Variable Formulation to Transient Conductive and Radiative Transport in Participating Media”, AIAA Journal of Thermophysics and Heat Transfer, vol. 9, #2, pp. 210-218 (1995).
67. LaClair, T., and J.I. Frankel, “Chebyshev Series Solution for Radiative Transport in a Medium with a Linearly Anisotropic Scattering Phase Function”, International Journal for Numerical Methods in Heat and Fluid Flow, vol. 5, #8, pp. 685-704 (1995).
68. Frankel, J.I., “A Note on the Integral Formulation of Kumar and Sloan”, Journal of Computational and Applied Mathematics, vol. 61, pp. 263-274 (1995).
69. Frankel, J.I., “Several Symbolic Augmented Chebyshev Expansions for Solving the Equation of Radiative Transfer”, Journal of Computational Physics, vol. 117, pp. 350-363 (1995).
70. Frankel, J.I., “A Galerkin Solution to a Regularized Cauchy Singular Integro-Differential Equation”, Quarterly of Applied Mathematics, vol. 53, #2, pp. 245-258 (1995).
71. Frankel, J.I., and S.R. Choudhury, “Some New Observations on the Classical Logistic Equation with Heredity”, Applied Mathematics and Computation, vol. 58, pp. 275-308 (1993).
72. Frankel, J.I., “Computational Attributes of the Integral Form of the Equation of Transfer”, Journal of Quantitative Spectroscopy and Radiative Transfer, vol. 46, #4, pp. 329-342 (1991).
73. Frankel, J.I., “A Regularized and Preconditioned Boundary Integral Solution to Heat Transfer in a Participating Gas Flow between Parallel Plates”, Numerical Heat Transfer, Part B, vol. 19, #1, pp. 105-126, (1991).
74. Frankel, J.I. and S.R. Sonikar, “Combined Radiative and Convective Heat Transfer of a Compressible, Transparent Gas through a Tube”, Comm. Int. J. Heat and Mass Transfer, vol. 17, #3, pp. 293-304 (1990).
75. Frankel, J.I., and J.J. Silvestri, “Radiative Exchange between Longitudinal Gray Fins”, J. Thermophysics and Heat Transfer, vol. 5, #1, pp. 120-122 (1991).
76. Frankel J.I., “Green’s Function Solution to Heat Transfer of a Transparent Gas through a Tube”,Numerical Heat Transfer, Part B, vol. 16, #2, pp. 231-243 (1989).
77. Frankel, J.I, “Nonlinear Heat Transfer: Solution of Singular, Nonlinear Integral Equations”, Engineering Analysis, vol. 8, #5, pp. 231-238 (1991).
78. Frankel J.I., and and T.P. Wang, “Radiative Exchange from an Array of Gray Fins using a Couple Integral Equation Formulation”, J. Thermophysics and Heat Transfer, vol. 2, #4, pp. 296-302 (1988).
79. Vick,B., J.H. Beale, and J.I. Frankel, “Integral Equation Solution for Internal Flow Subjected to a Variable Heat Transfer Coefficient”, J. Heat Transfer. vol. 109, pp.856-860 (1987).
80. Frankel J.I. and B.Vick, “An Exact Methodology for Solving Nonlinear Diffusion Equations Based on Integral Transforms”, Applied Numerical Mathematics, vol. 3, pp.467-477 (1987).
81. Frankel,J.I., B.Vick, and M.N. Ozisik,”General Formulation and Analysis of Hyperbolic Heat Conduction in Composite Media”, Int. J. Heat and Mass Transfer, vol. 30, #7, pp.1293-1305 (1987).
82. Frankel J.I., B.Vick,and M.N. Ozisik, “Flux Formulation in Hyperbolic Heat Conduction”, J. Applied Physics, vol. 58, #9, pp.3340-3345 (1985).
83. Thomas, J.R., J.I Frankel, and D.P.H. Hasselman, “Effect of Interface Reflections and Angle of Incidence of Radiation on Thermal Stresses in Semi-Transparent Material”, J. Thermal Stresses, vol.6, #1, pp.129-137 (1983).
84. Singh, J., J.I. Frankel J.R.Thomas,and D.P.H. Hasselman, “Thermal Stresses in Partially Absorbing Flat Plate Due to Sudden Interruption of the Steady State Asymmetric Radiation,II; Convective Cooling at Front Surface”, J. Thermal Stresses, vol. 6, #1, pp.25-34 (1983).
85. Frankel J.I., J.Singh, J.R.Thomas, and D.P.H. Hasselman, “Thermal Stresses in Partially Absorbing Flat Plate Due to Sudden Interruption of the Steady State Asymmetric Radiation, I; Convective Cooling at Rear Surface”, J. Thermal Stresses, vol. 6, #1, pp.15-24 (1983).