# Estimation of the Time-Dependent Heat Flux Using Temperature Distribution at a Point in a Three Layer System with None Homogeneous Boundary Conditions

Document Type : Original Article

Author

Islamic Azad University, Shahrood, Iran

Abstract

In this paper, the conjugate gradient method coupled with adjoint problem is used in order to solve the inverse heat conduction problem and estimation of the time- dependent heat flux using the temperature distribution at a point in a three layer system with none homogeneous boundary conditions. Also, the effect of noisy data on final solution is studied. For solving this problem the general coordinate method is used. The inverse heat conduction problem of estimating the transient heat flux, applied on part of the boundary of an irregular region is solved in this paper. The present formulation is general and can be applied to the solution of boundary inverse heat conduction problems over any region that can be mapped into a rectangle. The obtained results for few selected examples show the good accuracy of the presented method.  Also the solutions have good stability even if the input data includes noise. Applications of this model are in the thermal protect systems (t.p.s.) and heat shield systems.

Keywords

#### References

[1] Huang, C.H., Wang, P., "A three-dimensional inverse heat conduction problem in estimating surface heat flux by conjugate gradient method," International Journal of Heat and Mass Transfer, 1999, pp. 3387-3403.
[2]  Shiguemori, E.H., Harter, F.P., Campos Velho, H.F., da Silva, J.D.S., "Estimation of boundary conditions in conduction heat transfer by neural networks," Tendˆencias em Matem´atica Aplicada e Computacional, Vol. 3, No. 2,  2002, pp. 189-195.
[3] Volle, F., Maillet, D., Gradeck, M., Kouachi, A., Lebouché, M., "Practical application of inverse heat conduction for wall condition estimation on a rotating cylinder," International Journal of Heat and Mass Transfer, Vol. 52, 2009, pp. 210-221.
[4] Golbahar Haghighi, M.R., Eghtesad, M., Malekzadeh, P., Necsulescu,D.S., "Three-dimensional inverse transient heat transfer analysis of thick functionally graded plates," Energy Conversion and Management, Vol.  50, 2009, pp. 450-457.
[5] Su, J., Neto, A., "Two dimensional inverse heat conduction problem of source strength estimation in cylindrical rods," Applied Mathematical Modeling, Vol. 25, 2001, pp. 861- 872.
[6] Hsu, P.T., "Estimating the boundary condition in a 3D inverse hyperbolic heat conduction problem," Applied Mathematics and Computation, Vol. 177, 2006, pp. 453-464.
[7] Shi, J., Wang, J., "Inverse problem of estimating space and time dependent hot surface heat flux in transient transpiration cooling process," International Journal of Thermal Sciences, Vol. 48, 2009, pp. 1398-1404.
[8] Yang, Y. C., Chu, S. S., Chang, W. J., Wu, T. S., "Estimation of heat flux and temperature distributions in a composite strip and homogeneous foundation," International Communications in Heat and Mass Transfer, Vol.37, 2010, pp. 495-500.
[9] Wei, T., Li, Y.S., "An inverse boundary problem for one-dimensional heat equation with a multilayer domain," Engineering Analysis with Boundary Elements, Vol. 33, 2009, pp. 225-232.
[10] Chen, C. K., Su, C. R., "Inverse estimation for temperatures of outer surface and geometry of inner surface of furnace with two layer walls," Energy Conversion and Management, Vol. 49, 2008, pp. 301-310.
[11] Chen, T. C., Liu, C. C., Jang, H.Y., Tuan, P. C., "Inverse estimation of heat flux and temperature in multi-layer gun barrel," International Journal of Heat and Mass Transfer, Vol. 50, 2007, pp. 2060-2068.
[12] Haji-Sheikh, A., Beck, J.V., "Temperature solution in multi-dimensional multi-layer bodies," International Journal of Heat and Mass Transfer, Vol. 45, 2002, pp. 1865-1877.
[13] Ling, X., Atluril, S. N., "Stability analysis for inverse heat conduction problems," Tech Science Press., CMES, vol.13, No.3, 2006, pp. 219-228.
[14] Jiang, B.H., Nguyen, T.H., Prud’homme, M., "Control of the boundary heat flux during the heating process of a solid material," International Communications in Heat and Mass Transfer, 2005, Vol. 32, pp. 728-738.
[15] Jarny, Y., Ozisik, M.N., and Bardon, J.P., "A General Optimization Method Using Adjoint Equation for Solving Multidimensional Inverse Heat Conduction," Journal of Heat and Mass Transfer, Vol. 34, 1991, pp. 2911-2919.
[16] Daniel, J.W., Approximate Minimization of Functionals, Prentice-Hall Inc. Englewood Cliffs, 1971.
[17] Ozisik, M.N., Heat Conduction, 2nd ed., Wiley, New York, 1993.
[18] Alifanov,O.M., Inverse Heat Transfer Problems, Springer-Verlag, New York, 1994.
[19] Chen, S. G., Weng, C.I., Lin, J., "Inverse estimation of  transient temperature distribution in the end quenching test," Journal of Materials Processing Technology, Vol. 86, 1999, pp. 257-263.
[20] Ozisik, M. N., Orlando, R.B., Inverse Heat Transfer, New York, Taylor & Francis, 2000.
[21] Liu, F. B., "A hybrid method for the inverse heat transfer of estimating fluid thermal conductivity and heat capacity," International Journal of Thermal Sciences, Vol. 50 (5), 2011, pp. 718-724.
[22] Tai, B. L., Stephenson D. A.,  Shih, A. J., "An Inverse Heat Transfer Method for Determining Work piece Temperature in Minimum Quantity Lubrication Deep Hole Drilling," Journal of Manufacturing Science and Engineering, Vol.134 (2), 2012.
[23] Hong, F.J.,  Cheng, P.,  Wu, H.Y., Sun,  Z.,  "Evaporation/boiling heat transfer on capillary feed copper particle sintered porous wick at reduced pressure," International Journal of Heat and Mass Transfer, Vol. 63, 2013, pp. 389-400.
[24] Wu, T. S.,  Lee, H. L.,  Chang, W. J.,  Yang, Y. C., "An inverse hyperbolic heat conduction problem in estimating pulse heat flux with a dual-phase-lag model," International Communications in Heat and Mass Transfer, Vol. 60, 2015, pp.1-8.
[25] Beck, J., Black well, B., Clair, C. St., Inverse Heat Conduction,  J. Wiley, New York,1985.
[26] Mohammadiun, M., Rahimi, A.B., Khazaee, I., "Estimation of the time-dependent heat flux using temperature distribution at a point by conjugate gradient method," International Journal of Thermal Sciences, Vol.50, 2011, pp. 2443-2450.
[27] Rahimi, A.B., Mohammadiun, M., "Estimation of the Strength of the Time-dependent Heat Source Using Temperature Distribution at a Point in a Three Layer System," International Journal of Engineering, Vol. 25, 2012, pp. 343-351.
[28] Mohammadiun, H., Molavi, H., Talesh Bahrami, H.R., Mohammadiun, M., "Real-Time Evaluation of Severe Heat Load Over Moving Interface of Decomposing Composites," Journal of Heat Transfer, Vol. 134, 2012, pp. 111202,1-111202-7.
[29] Mohammadiun, M., Molavi, H., Talesh Bahrami, H.R., Mohammadiun, H., "Application of Sequential Function Specification Method in Heat Flux Monitoring of Receding Solid Surfaces," Heat Transfer Engineering, Vol. 35(10), 2014, pp. 933-941.