Determination of the time-dependent reaction coefficient and the heat flux in a nonlinear inverse heat conduction problem

Diffusion processes with reaction generated by a nonlinear source are commonly encountered in practical applications related to ignition, pyrolysis and polymerization. In such processes, determining the intensity of reaction in time is of crucial importance for control and monitoring purposes. Therefore, this paper is devoted to such an identification problem of determining the time-dependent coefficient of a nonlinear heat source together with the unknown heat flux at an inaccessible boundary of a one-dimensional slab from temperature measurements at two sensor locations in the context of nonlinear transient heat conduction. Local existence and uniqueness results for the inverse coefficient problem are proved when the first three derivatives of the nonlinear source term are Lipschitz continuous functions. Furthermore, the conjugate gradient method (CGM) for separately reconstructing the reaction coefficient and the heat flux is developed. The ill-posedness is overcome by using the discrepancy principle to stop the iteration procedure of CGM when the input data is contaminated with noise. Numerical results show that the inverse solutions are accurate and stable

Regularization of the semilinear sideways heat equation

A classical physical example of the sideways heat equation is represented by re-entry vehicles in the atmosphere where the temperature at the nozzle of a rocket is so high that any thermocouple attached to it would be destroyed. Instead one could measure both the temperature and heat flux, i.e. Cauchy data, at an interior boundary inward the capsule. In addition, we assume that there exists a heat source which is significantly dependent on space, time and temperature, and hence it cannot be neglected. This gives rise to a non-characteristic Cauchy inverse boundary value problem in the sense that the interior accessible boundary is overspecified, while the exterior hostile boundary is underspecified as nothing is prescribed on it. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the Cauchy data. In order to obtain a stable numerical solution, we propose two regularization methods to solve the semilinear problem in which the heat source is a Lipschitz function of temperature. We show rigourously, with error estimates provided, that the corresponding regularized solutions converge to the true solution strongly in L² uniformly with respect to the space coordinate under some a priori assumptions on the solution. These assumptions place no serious restrictions on the applicability of the results since in practice we always have some control and knowledge about how large the absolute temperature and heat flux are likely to be. Finally, in order to increase the significance of the study, numerical results are presented and discussed illustrating the theoretical findings in terms of accuracy and stability

An inverse problem for a one-dimensional time-fractional diffusion problem

Over the last two decades, anomalous diusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional dierential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. For example, the solution operator of the time fractional diusion diusion equation has only limited smoothing property, whereas the solution for the space fractional diusion equation may contain weakly singularity. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest. In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leer function and singular value decomposition, to examine the degree of ill-posedness of several \classical" inverse problems for fractional dierential equations involving a Djrbashian-Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order dierential equations. We discuss four inverse problems, i.e., backward fractional diusion, sideways problem, inverse source problem and inverse potential problem for time fractional diusion, and inverse Sturm-Liouville problem, Cauchy problem, backward fractional diusion and sideways problem for space fractional diusion. It is found that contrary to the wide belief, the in uence of anomalous diusion on the degree of ill-posedness is not denitive: it can either signicantly improve or worsen the conditioning of related inverse problems, depending crucially on the specic type of given data and quantity of interest. Further, the study exhibits distinct new features of \fractional" inverse problems, and a partial list of surprising observations is given below. (a) Classical backward diusion is exponentially ill-posed, whereas time fractional backward diusion is only mildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply that the latter always allows a more eective reconstruction. (b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but numerically can be nearly well-posed. (c) The classical Sturm-Liouville problem requires two pieces of spectral data to uniquely determine a general potential, but in the fractional case, one single Dirichlet spectrum may suce. (d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart, depending on the location of the lateral Cauchy data. In many cases, the precise mechanism of these surprising observations is unclear, and awaits further analytical and numerical exploration, which requires new mathematical tools and ingenuities. Further, our ndings indicate fractional diusion inverse problems also provide an excellent case study in the dierences between theoretical ill-conditioning involving domain and range norms and the numerical analysis of a nite-dimensional reconstruction procedure. Throughout we will also describe known analytical and numerical results in the literature

An inversion method for parabolic equations based on quasireversibility

AbstractThis paper is concerned with a new method to solve a linearized inverse problem for one-dimensional parabolic equations. The inverse problem seeks to recover the subsurface absorption coefficient function based on the measurements obtained at the boundary. The method considers a temporal interval during which time dependent measurements are provided. It linearizes the working equation around the system response for a background medium. It is then possible to relate the inverse problem of interest to an ill-posed boundary value problem for a differential-integral equation, whose solution is obtained by the method of quasireversibility. This approach leads to an iterative method. A number of numerical results are presented which indicate that a close estimate of the unknown function can be obtained based on the boundary measurements only

Novel Numerical Approaches for the Resolution of Direct and Inverse Heat Transfer Problems

This dissertation describes an innovative and robust global time approach which has been developed for the resolution of direct and inverse problems, specifically in the disciplines of radiation and conduction heat transfer. Direct problems are generally well-posed and readily lend themselves to standard and well-defined mathematical solution techniques. Inverse problems differ in the fact that they tend to be ill-posed in the sense of Hadamard, i.e., small perturbations in the input data can produce large variations and instabilities in the output. The stability problem is exacerbated by the use of discrete experimental data which may be subject to substantial measurement error. This tendency towards ill-posedness is the main difficulty in developing a suitable prediction algorithm for most inverse problems. Previous attempts to overcome the inherent instability have involved the utilization of smoothing techniques such as Tikhonov regularization and sequential function estimation (Beck’s future information method). As alternatives to the existing methodologies, two novel mathematical schemes are proposed. They are the Global Time Method (GTM) and the Function Decomposition Method (FDM). Both schemes are capable of rendering time and space in a global fashion thus resolving the temporal and spatial domains simultaneously. This process effectively treats time elliptically or as a fourth spatial dimension. AWeighted Residuals Method (WRM) is utilized in the mathematical formulation wherein the unknown function is approximated in terms of a finite series expansion. Regularization of the solution is achieved by retention of expansion terms as opposed to smoothing in the classical Tikhonov sense. In order to demonstrate the merit and flexibility of these approaches, the GTM and FDM have been applied to representative problems of direct and inverse heat transfer. Those chosen are a direct problem of radiative transport, a parameter estimation problem found in Differential Scanning Calorimetry (DSC) and an inverse heat conduction problem (IHCP). The IHCP is resolved for the cases of diagnostic deduction (discrete temperature data at the boundary) and thermal design (prescribed functional data at the boundary). Both methods are shown to provide excellent results for the conditions under which they were tested. Finally, a number of suggestions for future work are offered

Using the inverse heat conduction problem and thermography for the determination of local heat transfer coefficients and fin effectiveness for longitudinal fins

Heat transfer is a physical process in which energy is exchanged. It occurs in numerous applications, such as production of electricity, building climatisation, food preparation,... Since energy consumption has increased tremendously in the last decades and this trend will continue, the concept of energy efficiency has become omnipresent. In electronics miniaturization has become a trend. Desktops, laptops, dvd-players, mp3-players, televisions,... are getting thinner and/or smaller. Together with the increase in work speed and capacity, these small dimensions cause the energy density of electronic components (chips, processors,. . . ) to intensify significantly. As the electric power supply for these components is converted into heat, the component temperature rises. Hence, large amount of electricity are dissipated in a small surface area and cause high heat fluxes in the electronic components. To prevent overheating (and therefore failure) of electronic components, efficient heat removal is necessary. A cheap and almost universally applicable method for the cooling of electronics uses air as coolant in combination with a heat sink. The heat sinks are placed on the electronic component in order to distribute the heat and to create a better heat transfer. A heat sink mostly consists of longitudinal fins. Fin shape adjustments can improve the heat transfer, without the need for an increase in fin volume. This dissertation is specifically aimed at the research on longitudinal fins. It takes off looking for a measurement method to determine the performance of longitudinal fins as well as possible performance improvements by adjustments to these fins. The developed technique offers a global examination with a performance parameter. Moreover, it creates the possibility to study local heat transfer effects. In this work, the technique is applied to longitudinal fins, specifically fins for the cooling of electronics, but can be extended to other fin types. Chapter one also provides a summary of previous research on longitudinal fins. The number of studies on local heat transfer coefficients is limited and these studies are often inaccurate. A study of different fin performance indicators was also made, which indicated that the widely spread concept of fin efficiency is misleading, and a bad fin performance indicator. Nevertheless, many studies still aim for the highest possible fin efficiency, assuming this would guarantee the maximum heat transfer. A better, more reliable fin performance parameter is the fin effectiveness, or the performance ratio which is derived from it. As high fin effectiveness actually corresponds to a higher heat transfer, fin effectiveness was used as the fin performance indicator in this work. The developed measurement technique should not only be able to determine local heat transfer coefficients, it should also measure the fin effectiveness. To attain those goals, one has to determine the heat flux distribution in the fin. Normally, one does not measure heat fluxes, but temperatures, that make it possible to calculate the heat flux distribution. This requires a technique to accurately measure temperature profiles, and a numerical method to calculate the heat flux distribution from these measurements. This numerical method is developed in the second chapter. Determining heat fluxes from temperatures is known as the inverse heat conduction problem. This kind of problem is solved inversely. Whereas in a direct problem heat fluxes are imposed as boundary conditions and the temperature field is calculated from these conditions, in an inverse conduction problem the solution (temperature field) is known and the boundary conditions (heat fluxes) are determined from these temperatures. An introducing literature survey indicates that the inverse conduction problem is ill-posed and that it therefore can have several solutions. To obtain stable, physically correct solutions, mathematical methods are used. The second chapter offers a summary of the solution methods found in literature, which are all based on the minimization of a temperature functional. The inverse heat conduction problem studied in this work is three-dimensional, linear and steady state. Based on the summary of the different numerical techniques the most suitable methods are chosen. Two methods are taken into consideration: the steepest descent method (SDM) and the conjugate gradient method (CGM). Chapter two mathematically develops both of these similar techniques and writes the complete solution algorithm for both of them. These two solution algorithms are applied to some numerical test cases in chapter 3. The test cases consist of a rectangular longitudinal fin that partly covers a flat primary surface. Different heat transfer coefficient profiles are imposed on the fin walls and the primary surface. Using these boundary conditions, the temperature profiles on the same surfaces are calculated. These temperature profiles are considered as exact temperature measurements and are the boundary conditions for the inverse heat conduction problem. This inverse heat conduction problem is solved with both SDM and CGM. Afterwards, chapter three investigates the influence of measurement errors on the measured temperature profiles for two different measurement accuracies: 0.1°C and 0.5°C. Apparently SDM and CGM have a comparable accuracy, but CGM converges much faster. The introduction of measurement errors gives comparable results as in the ideal case of exact temperature measurements. Only at the edges the deviations increase significantly. Enlarging the measurement error from 0.1°C to 0.5°C does not lead to the expected drastic decrease in accuracy of the estimated profiles. The results are even comparable to the exact results. This indicates that the solution methods are not too sensitive to noise and thus suitable to process experimental measurement data. Relying on the results, CGM was chosen as solution method because of the faster convergence rate. Chapter 4 develops a measurement method using infrared thermography as measurement technique. Infrared thermography has the advantage that it is a noncontacting method. Thus the temperature field and measurement object are not disturbed by the measurement. Moreover, thermography makes it possible to get complete temperature profiles with a single measurement. The first part of the chapter explains some basic notions on radiation and thermography. Calibration methods are drawn up and applied. An error analysis is executed on the parameters that determine the incident radiation energy and on the camera specific properties, resulting in an uncertainty for the measured temperature values. The second part of the chapter explains the measurement setup. First the dimensions of the studied fins are determined based on the Reynolds analogy and on data from literature. Subsequently, the composition of the experimental setup is described. A low speed wind tunnel is used to set the environmental conditions and vary the Reynlods number (Re), which allows examining the influence of Re on the fin effectiveness and local heat transfer coefficients. A heat source is placed at the bottom of the fin, in combination with a guard heater to limit uncontrolled temperature losses. The power of the heat source is based on the fin temperature that should be attained to perform the most accurate temperature measurements with the infrared camera. The end of the chapter presents the different fin forms that will be studied: solid rectangular longitudinal fins and perforated fins with various numbers of perforations. The final chapter accomplishes the data reduction and presents the results. The temperature images, measured with the infrared camera during the experiments, are converted to a matrix with temperature values. This matrix can be used as a boundary condition for the inverse heat conduction problem that is solved with the developed solution method based on CGM. This solution makes it possible to determine the local heat fluxes and fin effectivenesss. The results obtained for the rectangular longitudinal fins agree with data from literature. The local heat transfer coefficients indicate the expected trends, and even show the influence of a horseshoe vortex at the base of the fin. The results for the perforated fins show the influence of the perforations and of restarting the boundary layer: after a perforation higher local heat transfer coefficients are found. The comparison with values from literature confirms the obtained results. The results for fin effectiveness are not accurate enough to draw conclusions for this. To conclude, chapter 6 presents the most important findings and perspectives for future work