Heat Transfer: A Problem Solving Approach

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Published New York ; London: Language English View all editions Prev Next edition 1 of 2. Check copyright status Cite this Title Heat transfer: Tariq Other Authors Grassie, Thomas. Physical Description xxx, p.

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Subjects Heat -- Transmission. Includes bibliographical references and index. Companion website contains additional Excel workbooks. View online Borrow Buy Freely available Show 0 more links A core task of engineers is to analyse energy related problems. The analytical treatment is usually based on principles of thermodynamics, fluid mechanics and heat transfer, but is increasingly being handled computationally.

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Such a norm can be written as where denotes the vector of unknown parameters and the superscript above denotes transpose. The vector is given by is real-valued bounded function defined on a closed bounded domain. The function may have many local minima in , but it has only one global minimum. When and have some attractive properties, for instance, being a differentiable concave function and being a convex region, then a local maximum problem can be solved explicitly by mathematical programming methods.

The Levenberg-Marquardt method, originally devised for application to nonlinear parameter estimation problems, has also been successfully applied to the solution of linear ill-conditioned problems. Such a method was first derived by Levenberg by modifying the ordinary least-squares norm. Later Marquardt derived basically the same technique by using a different approach. To minimize the least-squares norm 7 , we need to equate to zero the derivatives of with respect to each of the unknown parameters ; that is, Let us introduce the sensitivity or Jacobian matrix, as follows: For linear inverse problem, the sensitivity matrix is not a function of the unknown parameters.

Equation 12 can be solved then in explicit form as follows: In the case of a nonlinear inverse problem, the matrix has some functional dependence on the vector. The solution of 12 requires an iterative procedure, which is obtained by linearizing the vector with a Taylor series expansion around the current solution at iteration.

Such a linearization is given by where and are the estimated temperatures and the sensitivity matrix evaluated at iteration , respectively.

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Equation 14 is substituted into 13 and the resulting expression is rearranged to yield the following iterative procedure to obtain the vector of unknown parameters: The iterative procedure given by 15 is called the Gauss method. Such method is actually an approximation for the Newton or Newton-Raphson method. We note that 13 and the implementation of the iterative procedure given by 15 require the matrix to be nonsingular, or where is the determinant.

Formula 16 gives the so-called identifiability condition; that is, if the determinant of is zero, or even very small, the parameters , for , cannot be determined by using the iterative procedure of Problems satisfying are denoted as ill-conditioned. Inverse heat transfer problems are generally very ill-conditioned, especially near the initial guess used for the unknown parameters, creating difficulties in the application of 13 or The Levenberg-Marquardt method alleviates such difficulties by utilizing an iterative procedure in the form where is a positive scalar named damping parameter and is a diagonal matrix.

The purpose of the matrix term is to damp oscillations and instabilities due to the ill-conditioned character of the problem, by making its components large as compared to those of if necessary. With such an approach, the matrix is not required to be nonsingular in the beginning of iterations and the Levenberg-Marquardt method tends to the steepest descent method; that is, a very small step is taken in the negative gradient direction. The parameter is then gradually reduced as the iteration procedure advances to the solution of the parameter estimation problem, and then the Levenberg-Marquardt method tends to the Gauss method given by The following criteria were suggested in literature [ 13 ] to stop the iterative procedure of the Levenberg-Marquardt method given by The criterion given by 18 tests if the least-squares norm is sufficiently small, which is expected in the neighborhood of the solution for the problem.

Similarly, 19 checks if the norm of the gradient of is sufficiently small, since it is expected to vanish at the point where is minimum.

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The last criterion given by 20 results from the fact that changes in the vector of parameters are very small when the method has converged. Generally, these three stopping criteria need to be tested and the iterative procedure of the Levenberg-Marquardt method is stopped if any of them is satisfied. Different versions of the Levenberg-Marquardt method can be found in the literature, depending on the choice of the diagonal matrix and on the form chosen for the variation of the damping parameter.

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In this paper, we choose as Suppose that the vectors of temperature measurements are given at times , , and an initial guess is available for the vector of unknown parameters. Choose a value for , say , and. Then, consider the following. Solve the direct problem 1 — 3 with the available estimate in order to obtain the vector.

Compute the sensitivity matrix from 11 and then the matrix from 21 , by using the current value of.

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Solve the following linear system of algebraic equations, obtained from Compute the new estimate as. Solve the exact problem 1 — 3 with the new estimate in order to find. If , replace by and return to Step 4. If , accept the new estimate and emplace by. Check the stopping criteria given by Stop the iterative procedure if any of them is satisfied; otherwise, replace by and return to Step 3.