1Optimization and Development of Energy Technologies Division, Research Institute of Petroleum Industry (RIPI), Tehran, Iran
2School of Mechanical Engineering, Iran University of Science and Technology (IUST), Tehran, Iran
In the design of heat exchangers, it is necessary to determine the heat transfer rate between hot and cold fluids in order to calculate the overall heat transfer coefficient and the heat exchanger efficiency. Heat transfer rate can be determined by inverse methods. In this study, the unknown space-time dependent heat flux imposed on the wall of a heat exchanger internal tube is estimated by applying an inverse method and simulated temperature measurements at the specified points in the flow field. It is supposed that no prior information is available on the variation of the unknown heat flux function. Variable metric method which belongs to the function estimation approach is utilized to predicate the unknown function by minimizing an objective function. Four versions of the presented inverse method, named DFP, BFGS, SR1, and Biggs, are used to solve the problem and the results obtained by each version are compared. The estimation of the heat flux depends on the location of the sensor and the uncertainties associated with temperature measurements. The influence of each factor is investigated in this paper. Results show that variable metric method is a rapid and precise technique for estimating unknown boundary conditions in inverse heat convection problems.
 J. C. Bokar, and M. N. Ozisik, “An inverse analysis for estimation of time-varying inlet temperature in laminar flow inside a parallel plate duct”, International Journal of Heat and Mass Transfer, Vol. 38, No. 1, pp. 39-45, (1995).
 J. V. Beck, and K. A. Woodbury, “Inverse problems and parameter estimation: integration of measurements and analysis”, Measurement Science and Technology, Vol. 9, No. 6, pp. 839-847, (1998).
 M. N. Ozisik, and H. R. B. Orlande, Inverse Heat Transfer: Fundamentals and Applications, 1st ed., Taylor & Francis, New York, (2000).
 O. M. Alifanov, Inverse Heat Transfer Problems, 1st ed., Springer-Verlag, Berlin, (1994).
 L. Luksan, and E. Spedicato, “Variable metric methods for unconstrained optimization and nonlinear least squares”, Journal of Computational and Applied Mathematics, Vol. 124, No. 1-2, pp. 61-95, (2000).
 C. H. Huang, and M. N. Ozisik, “Inverse problem of determining unknown wall heat flux in laminar flow through a parallel plate duct”, Numerical Heat Transfer, Part A: Applications: An International Journal of Computation and Methodology, Vol. 21, No. 1, pp. 55-70, (1992).
 M. J. Colaco, and H. R. B. Orlande, “Inverse forced convection problem of simultaneous estimation of two boundary heat flux in irregularly shaped channels”, Numerical Heat Transfer, Part A: Applications, Vol. 39, No. 7, pp. 737-
 P. Ding, and W. Q. Tao, “Estimation of unknown boundary heat flux in laminar circular pipe flow using functional optimization approach: effects of Reynolds numbers,” Journal of Heat Transfer, Vol. 131, No. 2, pp. 1-9, (2009).
 C. H. Huang, I. C. Yuan, and H. Ay, “A three-dimensional inverse problem in imaging the local heat transfer coefficients for plate finned-tube heat exchangers”, International Journal of Heat and Mass Transfer, Vol. 46, No. 19, pp. 3629-3638, (2003).
 W. L. Chen, and Y. C. Yang, “An inverse problem in determining the heat transfer rate around two in line cylinders placed in a cross stream”, Energy Conversion and Management, Vol. 48, No. 7, pp. 1996-2005, (2007).
 F. Bozzoli, L. Cattani, C. Corradi, M. Mordacci, and S. Rainieri, “Inverse estimation of the local heat transfer coefficient in curved tubes: a numerical validation”, Journal of Physics: Conference Series, Vol. 501, No. 012002, (2014).
 J. H. Noha, W. G. Kima, K. U. Chab, and S. J. Yooka, “Inverse heat transfer analysis of multi-layered tube using thermal resistance network and Kalman filter”, International Journal of Heat and Mass Transfer, Vol. 89, pp.1016-1023, (2015).
 S .S. Rao, Optimization: Theory and Applications, 2nd ed., Wiley Eastern Limited, New Delhi, (1984).
 M. C. Biggs, “A note on minimization algorithms which make use of non-quadratic properties of the objective function”, IMA Journal of Applied Mathematics, Vol. 12, No. 3, pp. 337-338, (1973).