An inverse problem of triple-thickness parameters determination for thermal protective clothing with Stephan–Boltzmann interface conditions

  • 1 School of Mathematics, Shanghai University of Finance and Economics, 200433; and School of Sciences, Zhejiang Sci-Tech University, Hangzhou 310018, Shanghai, P. R. China
  • 2 School of Mathematics, Shanghai University of Finance and Economics, 200433, Shanghai, P. R. China
  • 3 Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia
  • 4 Department of Mathematics, Hokkaido University, Sapporo, Japan
  • 5 Institute of Textiles & Clothing, The Hong Kong Polytechnic University, and School of Architecture and Art, Central South University, Changsha 410075, Hong Kong, P. R. China
Tingyue Li, Sergey KabanikhinORCID iD:, Gen Nakamura, Faming WangORCID iD: and Dinghua XuORCID iD:


A seven-layers parabolic model with Stephan–Boltzmann interface conditions and Robin boundary conditions is mathematically formulated to describe the heat transfer process in environment-three layers clothing-air gap-body system. Based on this model, the solution to the corresponding inverse problem of simultaneous determination of triple fabric layers thickness is given in this paper, which satisfies the thermal safety requirements of human skin. By implementing a stable finite difference scheme, the thermal burn injuries on the skin of the body can be predicted. Then a kind of stochastic method, named as particle swarm optimization (PSO) algorithm, is developed to numerically solve the inverse problem. Numerical results indicate that the formulation of the model and proposed algorithm for solving the corresponding inverse problem are effective. Hence, the results in this paper will provide scientific supports for designing and manufacturing thermal protective clothing (TPC).

  • [1]

    D. Barr, W. Gregson and T. Reilly, The thermal ergonomics of firefighting reviewed, Appl. Ergonomics 41 (2010), 161–172.

    • Crossref
    • Export Citation
  • [2]

    M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, Inc, New Jersey, 2005.

  • [3]

    A. Cheng and H. Wang, An error estimate on a Galerkin method for modeling heat and moisture transfer in fibrous insulation, Numer. Methods for Partial Differential Equations 2 (2010), 504–517.

  • [4]

    P. Chitrphiromsri and A. V. Kuznetsov, Modeling heat and moisture transport in firefighter protective clothing during flash fire exposure, Heat Mass Transf. 41 (2005), 206–215.

  • [5]

    J. T. Fan, X. Y. Cheng and W. W. Sun, An improved model of heat and moisture transfer with phase change and mobile condensates in fibrous insulation and comparison with experimental results, Int. J. Heat Mass Transf. 47 (2004), 2343–2352.

    • Crossref
    • Export Citation
  • [6]

    J. T. Fan, Z. X. Luo and Y. Li, Heat and moisture transfer with sorption and condensation in porous clothing assemblies and numerical simulation, Int. J. Heat Mass Transf. 43 (2000), 2989–3000.

    • Crossref
    • Export Citation
  • [7]

    J. T. Fan and X. H. Wei, Heat and moisture transfer through fibrous insulation with phase change and mobile condensates, Int. J. Heat Mass Transf. 19 (2002), 4045–4055.

  • [8]

    B. Farnworth, Mechanisms of heat flow through clothing insulation, Textile Res. J. 53 (1983), 717–725.

    • Crossref
    • Export Citation
  • [9]

    A. Ghazy, Influence of thermal shrinkage on protective clothing performance during fire exposure: Numerical investigation, Mech. Eng. Res. 4 (2014), 1–15.

  • [10]

    A. Ghazy and D. J. Bergstrom, Influence of the air gap between protective clothing and skin on clothing performance during flash fire exposure, Heat Mass Transf. 47 (2011), 1275–1288.

    • Crossref
    • Export Citation
  • [11]

    A. Ghazy and D. J. Bergstrom, Numerical simulation of heat transfer in firefighters’ protective clothing with multiple air gaps during flash fire exposure, Numer. Heat Transf. 61 (2012), 569–593.

    • Crossref
    • Export Citation
  • [12]

    A. Ghazy and D. J. Bergstrom, Numerical simulation of the influence of fabric’s motion on protective clothing performance during flash fire exposure, Heat Mass Transf. 49 (2013), 775–788.

    • Crossref
    • Export Citation
  • [13]

    P. W. Gibson, Multiphase heat and mass transfer through hygroscopic porous media with applications to clothing materials, Fiber. 53 (1996), 183–194.

  • [14]

    F. C. Henriques and A. R. Moritz, Studies of thermal injuries I: The conduction of heat to, and through skin, and the temperatures attained therein, a theoretical and experimental investigation, Amer. J. Pathol. 23 (1947), 531–549.

  • [15]

    J. R. Lawson, W. D. Walton, N. P. Bryner and F. K. Amon, Estimates of thermal properties for fire fighters’ protective clothing materials, preprint (2005).

  • [16]

    W. E. Mell and J. R. Lawson, A heat transfer model for firefighters’ protective clothing, Fire Technol. 36 (2000), 39–68.

    • Crossref
    • Export Citation
  • [17]

    M. F. Modest, Radiative Heat transfer, 2nd ed., Academic Press, Boston, 2003.

  • [18]

    G. Song, Modeling thermal protection outfits for fire exposures, Ph.D. thesis, North Carolina State University, 2003.

  • [19]

    G. W. Song, R. L. Barker, H. Hamouda, A. V. Kuznetsov, P. Chitrphiromsri and R. V. Grimes, Modeling the thermal protective performance of heat resistant garments in flash fire exposures, Textile Res. J. 74 (2004), 1033–1040.

    • Crossref
    • Export Citation
  • [20]

    G. W. Song, P. Chitrphiromsri and D. Ding, Numerical simulations of heat and moisure transport in thermal protective clothing under flash fire conditions, Int. J. Occupational Safety & Ergonomics Jose 14 (2008), 89–106.

    • Crossref
    • Export Citation
  • [21]

    A. M. Stoll and M. A. Chianta, Method and rating system for evaluation of thermal protection, Aerospace Medicine 11 (1969), 1232–1238.

  • [22]

    Y. Su, J. Z. He and J. Li, An improved model to analyze radiative heat transfer in flame-resistant fabrics exposed to low-level radiation, Textile Res. J. 16 (2016), 1953–1967.

  • [23]

    D. A. Torvi and J. D. Dale, Heat transfer in thin fibrous materials under high heat flux, Fire Technol. 35 (1999), 210–231.

    • Crossref
    • Export Citation
  • [24]

    D. A. Torvi, D. J. Douglas and B. Faulkner, Influence of air gaps on bench-top test results of flame resistant fabrics, J. Fire Protection Eng. 10 (1999), 1–12.

    • Crossref
    • Export Citation
  • [25]

    Udayraj and F. Wang, A three-dimensional conjugate heat transfer model for thermal protective clothing, Int. J. Thermal Sci. 130 (2018), 28–46.

    • Crossref
    • Export Citation
  • [26]

    J. Vershoor and P. Greebler, Heat transfer by gas conduction and radiation in fibrous insulation, Trans. Amer. Math. Soc. Mech. Eng. 74 (1952), 961–968.

  • [27]

    D. H. Xu, Mathemtical Modeling of Heat and Moisture Transfer within Textiles and Corresponding Inverse Problems of Textile Material Design, Science Press, Beijing, 2014.

  • [28]

    D. H. Xu, R. L. Chen and M. B. Ge, Inverse problems of textile material design based on comfort of clothing, Commun. Appl. Comput. Math. 3 (2012), 332–341.

  • [29]

    D. H. Xu, Y. B. Chen and X. H. Zhou, Type design for textile materials under low temperature: Modeling, numerical algorithm and simulation, Int. J Heat Mass Transf. 60 (2013), 582–590.

    • Crossref
    • Export Citation
  • [30]

    D. H. Xu, J. X. Cheng, Y. B. Chen and M. B. Ge, An inverse problem of thickness design for bilayer textile materials under low temperature, J. Phys. Conf. Ser. 290 (2011), Article ID 12018.

  • [31]

    D. H. Xu and M. B. Ge, Thickness determination in textile material design: dynamic modeling and numerical algorithms, Inverse Problems 28 (2012), Article ID 035011.

  • [32]

    D. H. Xu, L. Wen and B. Xu, An inverse problems of bilayer textile thickness determination in dynamic heat and moisture transfer, Appl. Anal. 93 (2013), 445–465.

  • [33]

    Y. H. Xu, D. H. Xu, L. P. Zhang and X. H. Zhou, A new inverse problem for the determination of textile fabrics, Inverse Probl. Sci. Eng. 23 (2015), 635–650.

    • Crossref
    • Export Citation
  • [34]

    G. F. Yang, M. Yamamoto and J. Cheng, Heat transfer in composite materials with Stefan–Boltzmann interface conditions, Math. Methods Appl. Sci. 11 (2010), 1297–1314.

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This journal presents original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published.