Inverse Estimation of Time-Dependent Heat Flux in Stagnation Region of Annular Jet on a Cylinder Using Levenberg–Marquardt Method

Document Type : Research Article

Authors

1 Department of Mechanical Engineering, Shahrood Branch, Islamic Azad University, Shahrood, I.R. IRAN

2 Department of Electrical Engineering, Shahrood Branch, Islamic Azad University, Shahrood, I.R. IRAN

Abstract

All solving methods available in the literature are formulated for direct solution of stagnation point flow and its heat transfer impinging on the surfaces with known boundary conditions. In this study for the first time, an anumerical code based on Levenberg–Marquardt method is presented for solving the inverse heat transfer problem of an annular jet on a cylinder and estimating the time-dependent heat flux using temperature distribution at a specific point. Also, the effect of noisy data on the final results is studied. For this purpose, the numerical solution of the dimensionless temperature and the convective heat transfer in a radial incompressible flow on a cylinder rod is carried out as a direct problem.In the direct problem, the free stream is steady with an initial flow strain rate of k. Using similarity variables and appropriate transformations, momentum and energy equations are converted into Semi-similar equations. The new equation systems are then discretized using an implicit finite difference method and solved by applying the Tri-Diagonal Matrix Algorithm (TDMA). The heat flux is then estimated by applying the Levenberg–Marquardt parameter estimation approach. This technique is an iterative approach based on minimizing the least-square summation of the error values, the error being the difference between the estimated and measured temperatures. Results of the inverse analysis indicate that the Levenberg–Marquardt algorithm is an efficient and acceptably stable technique for estimating heat flux in axisymmetric stagnation flow. This method also exhibits considerable stability for noisy input data. The maximum value of the sensitivity coefficient is related to the estimation of exponential heat flux and its value is 0.1619 also the minimum value of the sensitivity coefficient is 5.62´10-6 which is related to the triangular heat flux. The results show that the parameter estimation error in calculating the triangular and trapezoidal heat flux is greater than the exponential and sinus–cosines heat flux because the maximum value of RMS error is obtained for these two cases, which are 0.481 and 0.489, respectively the reason for the increase in the errors in estimating these functions is the existence of points where the first derivative of the function does not exist. The problem is particularly important in pressure-lubricated bearings. 

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Main Subjects

References

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