Parameter Estimation on a Model of a Heat-Conducting Rod: Mathematical Analysis and Numerical Computations
Ma. Cristina R. Bargo, Ricardo C.H. del Rosario*, and Jose Ernie C. Lope
Institute of Mathematics
University of the Philippines Diliman
C.P. Garcia St., Diliman 1101
Quezon City, Philippines
We model the heat flow of an experiment involving a metallic rod with a heat source at one end. By fitting the mathematical model to experimental data, we were able to estimate the ratios of the rod's thermal conductivity, heat capacity, and flux of the heat source over the heat transfer coefficient. Temperature measurements were obtained via thermocouples located at seven points along the length of the rod. Both steady-state and time-dependent data were used in the parameter estimation, and for the time-dependent case, a numerical algorithm for approximating the partial differential equation was implemented. Since the model featuring realistic boundary conditions does not have an explicit solution, we showed the well-posedness of the model using mathematical analysis. The convergence of the finite-dimensional approximations of the model to the infinite-dimensional solution was also discussed. To solve the optimization problem arising from the parameter estimation problem, we used the least squares and genetic algorithms. Our numerical results indicate that these two optimization algorithms converged to the same solution.
In this paper, we use the one-dimensional heat equation to model an actual experiment consisting of a metallic rod with a heat source at one end, and with thermocouples measuring the transient and steady-state temperature. The partial differential equation was formulated to incorporate heat loss along the length of the rod and to satisfy realistic boundary conditions: heat flux input at one end and heat loss at the other end. The resulting equation has no analytic solution due to the incorporation of realistic boundary conditions, hence numerical methods must be implemented to approximate the solution. Thus, the convergence of the numerical approximations must be established and to accomplish this, we first show the existence and uniqueness of the partial differential equation posed in an infinite dimensional Hilbert space, then we cite existing results that guarantee the convergence of the numerical approximations to the infinite-dimensional solution.
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