Prof. Tensi discovered four types of heat transfer modes (HTM) which should be taken into account when solving inverse problem (IP). Theoretically, the same cooling curve of standard probe can be generated by each single mode. Boundary condition for each HTM is quite different. To solve correctly IP, one should know exactly what kind of HTM takes place during quenching.
If the solution of the Inverse Problem for each mode exists already and if the obtained cooling curves are rigourously the same, only a physical appreciation based on the definition of the boundary conditions and of the all thermal model can distinguish the more available. Moreover, in a general way, for an Inverse problem, the uniqueness of the solution is still an open mathematical problem.
If the cooling curves have small quantitative diferences, the more adequate HTM model is those which made the best fit of the measured data. In this case it is neceassy to use the same metrics to estimate the error between the model's predictions and the measurements.
If the solution of the Inverse Problem for each mode exists already and if the obtained cooling curves are rigourously the same, only a physical appreciation based on the definition of the boundary conditions and of the all thermal model can distinguish the more available. Moreover, in a general way, for an Inverse Problem, the uniqueness of the solution is still an open mathematical problem.
If the cooling curves have small quantitative diferences, the more adequate HTM model is those which made the best fit of the measured data. In this case it is necessary to use the same metrics to estimate the error between the model's predictions and the measurements.
@Nicolai Kobasco.
For a heat transfer problem described by parabolic formulation (heat transfer by waves can be formulated by a hyperbolic equation), different initial conditions are going to (exponentially) collapse at the same profile, after certain period of time. Therefore, small differences in the measured data (below to the measuring equipment error) imply to be impossible to identify the correct initial condition. The way to overcome such trouble is to impose some additional information (for example: request a smoothest solution, with the best fitting to the observations = measurements).
From a mathematical point of view, the parabolic heat equation has a compact operator associated to it. The spectrum of such operator has the zero as as eigenvalue, if the space dimension is infinity (this is a fundamental result from functional analysis = spectral theory of operators). Therefore, there is no formal inverse for such operator. For practical applications, this means: we can not compute a correct solution to estimate initial condition (or boundary condition, or source terms, or model parameters) from measurements. It is necessary to modify the operator (removing its singularity, adding a non-zero diagonal operator, for example) in order to compute a good (inverse) solution.
Dear Colleagues,
Thank you very much for your comments and good advice. I completely agree with you.
We should consider hyperbolic heat conductivity equation instead of parabolic and specific boundary condition for each mode. Some additional information explaining types of heat transfer modes is enclosed.
Let's follow each other to discuss further this problem.
Nikolai Kobasko
And to close my answer, a class of regularization techniques exists for numerical algorithm used to solve an inverse problem (add of a diagonal matrix to the initial ones, with small controled diagonal values, matrix conditionning method to diminuish the conditionning number equal to the rapport between the maximal eigenvalues and the minimal ones ) can allows to a correct solution.
Who can formulate boundary condition for the first type of boiling process summarized in the attachment?
The solution of inverse problems can be classified into two methods. The first one is based on a mathematical model of the respective physical phenomenon (so-called phenomenological methods). The second one is the methods based on the algorithms for the analysis of digital signals (so-called algorithmic methods). The phenomenological methods involving a mathematical model assume that the configuration of the system is varied until the norm of the mismatch between the model solution and the experimentally obtained signal is minimized. A good result is only guaranteed if the physics of the phenomenon in the model is close to reality. In algorithmic methods, the inversion procedure applied to experimental data is considered as an image-recognition problem. In this case, the signal is identified as a representative of the classes associated with known of the system types. The classification algorithms, which are most frequently used, are developed through identification of diagnostic signatures. This approach assumes the use of an artificial neural network (or another optimization algorithm) trained with the signals from a predefined database. In this case the solution is generally guaranteed.
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