Extending Inverse Heat Conduction Method to Estimate Flight Trajectory of a Reentry Capsule

Document Type : Research Article


1 Amirkabir University of Technology, 15875-4413 Tehran, Iran

2 صنعتی امیرکبیر*مهندسی هوافضا


This study is dedicated to the solution of the inverse heat conduction problem for estimation of the time-varying velocity and altitude profiles regarding the flight trajectory of an earth-entry capsule. Four tungsten-rhenium sensors are supposed to be embedded inside the ablative heat shield of the probe, three of which cannot tolerate high-temperature conditions and burn out during entry and the other one remained intact until the end of the simulation. The conventional Levenberg-Marquardt method is reinforced by a relaxation scheme to prevent unfavorable severe oscillations encountered in the inverse iterations. To keep the generality of the method, no prior knowledge on the thermal condition and surface recession of ablative insulator is utilized in the current estimation. Therefore, in the associated direct problem, velocity and altitude profiles are given and the temperature field inside the heat shield is determined. Accordingly, a solution of the direct problem consists of (i) bow shock calculations in dissociated air (ii) boundary layer solver to compute stagnation heating rate (iii) identification of the thermal response of charring ablative heat shield. It is shown that if the standard deviation of the temperature measurement error is 5 K, estimation of altitude and velocity are associated with approximately 10 and 5 percent normalized error, respectively.


Main Subjects

[1] O.M. Alifanov, Inverse heat transfer problems, Springer Science & Business Media, 2012.
[2] J.V. Beck, K.J. Arnold, Parameter estimation in engineering and science, James Beck, 1977.
[3] M.N. Ozisik, Inverse heat transfer: fundamentals and applications, Routledge, 2018.
[4] F. Bozzoli, A. Mocerino, S. Rainieri, P. Vocale, Inverse heat transfer modeling applied to the estimation of the apparent thermal conductivity of an intumescent fire retardant paint, Experimental Thermal and Fluid Science, 90 (2018) 143-152.
[5] C.-W. Chang, C.-H. Liu, C.-C. Wang, Review of computational schemes in inverse heat conduction problems, Smart Science, 6(1) (2018) 94-103.
[6] O. Fabela, S. Patil, S. Chintamani, B.H. Dennis, Estimation of effective thermal conductivity of porous Media utilizing inverse heat transfer analysis on cylindrical configuration, in:  ASME 2017 International Mechanical Engineering Congress and Exposition, American Society of Mechanical Engineers Digital Collection, 2017.
[7] M. Jahedi, F. Berntsson, J. Wren, B. Moshfegh, Transient inverse heat conduction problem of quenching a hollow cylinder by one row of water jets, International Journal of Heat and Mass Transfer, 117 (2018) 748-756.
[8] F. Kowsary, A. Behbahaninia, A. Pourshaghaghy, Transient heat flux function estimation utilizing the variable metric method, International communications in heat and mass transfer, 33(6) (2006) 800-810.
[9] T. Loulou, Combined parameter and function estimation with application to thermal conductivity and surface heat flux, Journal of Heat Transfer, 129(10) (2007) 1309-1320.
[10] M. Mohammadiun, Time-dependent heat flux estimation in multi-layer systems by inverse method, Journal of Thermophysics and Heat Transfer, (null) (2016) 599-607.
[11] H. Molavi, A. Hakkaki-Fard, R.K. Rahmani, A. Ayasoufi, M. Molavi, A novel methodology for combined parameter and function estimation problems, Journal of Heat Transfer, 132(12) (2010) 121301.
[12] C.-y. Yang, Estimation of boundary conditions in nonlinear inverse heat conduction problems, Journal of thermophysics and heat transfer, 17(3) (2003) 389-395.
[13] J. Zueco, F. Alhama, C.G. Fernandez, Numerical nonlinear inverse problem of determining wall heat flux, Heat and mass transfer, 41(5) (2005) 411-418.
[14] K. Dowding, J. Beck, A. Ulbrich, B. Blackwell, J. Hayes, Estimation of thermal properties and surface heat flux in carbon-carbon composite, Journal of Thermophysics and Heat Transfer, 9(2) (1995) 345-351.
[15] V. Petrushevsky, S. Cohen, Nonlinear inverse heat conduction with a moving boundary: heat flux and surface recession estimation,  (1999).
[16] A.P. de Oliveira, H.R. Orlande, Estimation of the heat flux at the surface of ablating materials by using temperature and surface position measurements, Inverse Problems in Science and Engineering, 12(5) (2004) 563-577.
[17] A. Hakkaki-Fard, F. Kowsary, Heat flux estimation in a charring ablator, Numerical Heat Transfer, Part A: Applications, 53(5) (2007) 543-560.
[18] H.B. Khaniki, S.H. Karimian, Determining the heat flux absorbed by satellite surfaces with temperature data, Journal of Mechanical Science and Technology, 28(6) (2014) 2393-2398.
[19] H. Mohammadiun, H. Molavi, H.R.T. Bahrami, M. Mohammadiun, Real-Time Evaluation of Severe Heat Load Over Moving Interface of Decomposing Composites, Journal of Heat Transfer, 134(11) (2012) 111202.
[20] M. Mohammadiun, H. Molavi, H.R.T. Bahrami, H. Mohammadiun, Application of sequential function specification method in heat flux monitoring of receding solid surfaces, Heat Transfer Engineering, 35(10) (2014) 933-941.
[21] H. Molavi, A. Hakkaki-Fard, M. Molavi, R.K. Rahmani, A. Ayasoufi, S. Noori, Estimation of boundary conditions in the presence of unknown moving boundary caused by ablation, International Journal of Heat and Mass Transfer, 54(5-6) (2011) 1030-1038.
[22] H. Molavi, I. Pourshaban, A. Hakkaki-Fard, M. Mohlavi, Ablative Materials' Boundary Condition Simulation by Applying Inverse Approach and Euler Solver, Journal of Thermophysics and Heat Transfer, 26(1) (2012) 47-56.
[23] H. Molavi, R.K. Rahmani, A. Pourshaghaghy, E.S. Tashnizi, A. Hakkaki-Fard, Heat flux estimation in a nonlinear inverse heat conduction problem with moving boundary, Journal of Heat Transfer, 132(8) (2010) 081301.
[24] A. Plotkowski, M.J.M. Krane, The use of inverse heat conduction models for estimation of transient surface heat flux in electroslag remelting, Journal of Heat Transfer, 137(3) (2015) 031301.
[25] T.-S. Wu, H.-L. Lee, W.-J. Chang, Y.-C. Yang, An inverse hyperbolic heat conduction problem in estimating pulse heat flux with a dual-phase-lag model, International Communications in Heat and Mass Transfer, 60 (2015) 1-8.
[26] W.D. Henline, M.E. Tauber, Trajectory-based heating analysis for the European Space Agency/Rosetta Earth return vehicle, Journal of Spacecraft and Rockets, 31(3) (1994) 421-428.
[27] J.A. Fay, F.R. Riddell, Theory of stagnation point heat transfer in dissociated air, Journal of the Aerospace Sciences, 25(2) (1958) 73-85.
[28] M.E. Tauber, K. Sutton, Stagnation-point radiative heating relations for Earth and Mars entries, Journal of Spacecraft and Rockets, 28(1) (1991) 40-42.
[29] C.B. Moyer, R.A. Rindal, An analysis of the coupled chemically reacting boundary layer and charring ablator. part 2-finite difference solution for the in-depth response of charring materials considering surface chemical and energy balances,  (1968).
[30] C.F. Hansen, Approximations for the thermodynamic and transport properties of high-temperature air, National Aeronautics and Space Administration, 1959.
[31] H. Molavi, I. Pourshaban, A. Hakkaki-Fard, M. Molavi, A. Ayasoufi, R.K. Rahmani, Inverse identification of thermal properties of charring ablators, Numerical Heat Transfer, Part B: Fundamentals, 56(6) (2010) 478-501.
[32] R. POTTS, Hybrid integral/quasi-steady solution of charring ablation, in:  5th Joint Thermophysics and Heat Transfer Conference, 1990, pp. 1677.
[33] S. Williams, D.M. Curry, Thermal protection materials: thermophysical property data,  (1992).
[34] V. Tahmasbi, S. Noori, Thermal analysis of honeycomb sandwich panels as substrate of ablative heat shield, Journal of Thermophysics and Heat Transfer, 32(1) (2017) 129-140.