Final Report Summary - HTC (Determination of Heat Transfer Coefficients by Inverse Methods)
The inverse problem methodology is referred to as one of the most dynamically developing branches of modern science having multiple and diverse applications in industry. So, why did our solution of inverse heat transfer coefficient problems become necessary and what will account in the near future the rapid development of this area of research? The principal reason for specific interest to these problems is made up of the practical industrial requirements that usually encounter unsteadiness, nonlinearity and correlation of processes. These effects severely limit the application of classical experimental methods and demand the formulation of new approaches such as ours, as well as the development of new numerical methods for inverse problem solving. The main advantage of these methods is that they enable us to conduct posterior experimental studies under natural and realistic conditions or, directly in action by the engineering system. Our new formulations/models also have produced a larger amount of information and insight, so that, as an immediate impact, experimental studies can proceed more rapidly and less costly than in traditional methods of trial and error. Furthermore, our study contributes to the improvement and competitiveness of the EU industry through the adoption of more effective and efficient computational techniques, whilst adding rigour to the state of the art of research in the field of inverse problems.
The research performed in this project will be relevant, important and of interest to aerospace and nuclear industries in which inverse problem formulations are often adopted. Furthermore, the determination of thermal boundary coefficients which were investigated as objectives of our project, constitutes important inverse problems in design and optimization processes in heat transfer and polymer industries. For example, the knowledge of the ambient temperature - which refers to the temperature surrounding a heated or cooled specimen – is very important for safe and efficient performance of heat transfer equipment, e.g. thermal flow sensors. Moreover, as our project proposes, by allowing the ambient temperature to depend on space or time we obtain more realistic models for heat transfer in building enclosures, e.g. glazed surfaces where it is non-uniform depending on the local air patterns such as the type of flow and the external weather conditions. In another example, the heat transfer across the surface of a heat conducting body is accompanied by a heat transfer coefficient whose knowledge is crucial in characterising the contribution that this surface makes to the overall thermal resistance – the inverse of the thermal contact conductance – of the system. Improved understanding and characterisation capability of the heat transfer coefficient is very important in polymer processing as it can lead to reductions in scrap rates due to the avoidance of warpage and shrinkage of the product.
In the project, we have investigated determining the ambient temperature (linear problem) or the surface heat transfer coefficients (nonlinear problem) which may depend on space, time or temperature. This is modelled as a linkage function in a linear convective or, nonlinear Robin boundary condition of the third kind. At the steady state, the problem is more known as corrosion coefficient identification and it can be reduced to solving a Cauchy problem for an elliptic equation. However, the unsteady case which we propose is more useful since the extra time dimension offers a wider variety of inverse problem formulations. This in turn, will offer to practice a wider range of validation and verification procedures for material testing under harsh and hostile conditions commonly encountered in high pressures/high temperature environments in the aerospace and nuclear industries.
The additional information sufficient to guarantee a unique solution may come in the form of a point, instant or a boundary integral measurement of the temperature. In our analyses, both classical and weak solutions were sought. Although the existence and uniqueness of the solution may hold under certain restrictions on the input data, the inverse problems are still ill-posed since small errors, which are inherently present in any practical measurement, cause large errors in the output desired solution – in our case the unknown thermal boundary coefficients. Consequently, in order to obtain a stable and physically realistic solution regularization needs to be incorporated. In our project this has been achieved by employing the conjugate gradient method (CGM) for minimizing the penalised Tikhonov regularization functional. New formulae for the gradients of the objective functionals are delivered. When the regularization parameter is set to zero, the semi-convergent iterative algorithm is stopped according to the discrepancy principle.
Another, important and useful feature of our mathematical and numerical models is that they are multi-dimensional. As far as the computational efficiency is concerned, since the governing heat equation that we consider is linear with constant thermal property coefficients, the boundary element method (BEM) that we adopt is the most appropriate numerical method for discretising the direct and adjoint problems involved in the CGM algorithm. Moreover, all the knowns (input) and unknowns (output) in the inverse problems that we consider are at the boundary, and the discretisation of the boundary only is the essence of the BEM.
The variational problems have been discretised by the BEM and convergence results of the CGM have been proved which add rigour to our investigation. The fellow has served the purpose of transferring knowledge by bringing his unique and solid mathematical background from the outside the EU into the EU. Tested and verified on a wide variety of benchmark examples, the numerically obtained results showed to be accurate, stable and robust, and ready for practical validation.
The research developed in this project will allow to broaden largely the information on the thermal state of structures used and newly designed and will give a possibility to introduce new structures and products, and bring new power-consumable processes to a commercial status.
The fellowship has allowed the fellow to build substantially on his research experience in inverse heat transfer problems and to expose and immerse himself in a vigorous, demanding and active research atmosphere in which collaboration between scientists and engineers is the norm.
The mathematical knowledge that the fellow has brought into Europe on the theoretical analysis of the inversely formulated heat transfer coefficient identification problems has helped making Europe more competitive. The open issues that were addressed and solved in this research proposal make a substantial impact and contribute to the increasing level of fundamental research activity being performed in the inverse theory and numerics in Europe.
The fellowship has facilitated a lasting relationship between the researchers at Leeds and the fellow that will enable the knowledge transfer process to continue beyond the life of the fellowship. The success of this research provides significant additional momentum to develop the theme of determining heat transfer coefficients into a broader investigation encompassing the inverse mathematical modelling of inverse problems and its related computational implementation. As a continuation of this project which both the scientist-in-charge and the fellow envisage, future work will consist in extending some of the inverse analysis performed in this project to composite heat conductors involving the determination of the thermal contact conductance during metal casting, of exhaust gases, and in plate finned-tube heat exchangers.
The research performed in this project will be relevant, important and of interest to aerospace and nuclear industries in which inverse problem formulations are often adopted. Furthermore, the determination of thermal boundary coefficients which were investigated as objectives of our project, constitutes important inverse problems in design and optimization processes in heat transfer and polymer industries. For example, the knowledge of the ambient temperature - which refers to the temperature surrounding a heated or cooled specimen – is very important for safe and efficient performance of heat transfer equipment, e.g. thermal flow sensors. Moreover, as our project proposes, by allowing the ambient temperature to depend on space or time we obtain more realistic models for heat transfer in building enclosures, e.g. glazed surfaces where it is non-uniform depending on the local air patterns such as the type of flow and the external weather conditions. In another example, the heat transfer across the surface of a heat conducting body is accompanied by a heat transfer coefficient whose knowledge is crucial in characterising the contribution that this surface makes to the overall thermal resistance – the inverse of the thermal contact conductance – of the system. Improved understanding and characterisation capability of the heat transfer coefficient is very important in polymer processing as it can lead to reductions in scrap rates due to the avoidance of warpage and shrinkage of the product.
In the project, we have investigated determining the ambient temperature (linear problem) or the surface heat transfer coefficients (nonlinear problem) which may depend on space, time or temperature. This is modelled as a linkage function in a linear convective or, nonlinear Robin boundary condition of the third kind. At the steady state, the problem is more known as corrosion coefficient identification and it can be reduced to solving a Cauchy problem for an elliptic equation. However, the unsteady case which we propose is more useful since the extra time dimension offers a wider variety of inverse problem formulations. This in turn, will offer to practice a wider range of validation and verification procedures for material testing under harsh and hostile conditions commonly encountered in high pressures/high temperature environments in the aerospace and nuclear industries.
The additional information sufficient to guarantee a unique solution may come in the form of a point, instant or a boundary integral measurement of the temperature. In our analyses, both classical and weak solutions were sought. Although the existence and uniqueness of the solution may hold under certain restrictions on the input data, the inverse problems are still ill-posed since small errors, which are inherently present in any practical measurement, cause large errors in the output desired solution – in our case the unknown thermal boundary coefficients. Consequently, in order to obtain a stable and physically realistic solution regularization needs to be incorporated. In our project this has been achieved by employing the conjugate gradient method (CGM) for minimizing the penalised Tikhonov regularization functional. New formulae for the gradients of the objective functionals are delivered. When the regularization parameter is set to zero, the semi-convergent iterative algorithm is stopped according to the discrepancy principle.
Another, important and useful feature of our mathematical and numerical models is that they are multi-dimensional. As far as the computational efficiency is concerned, since the governing heat equation that we consider is linear with constant thermal property coefficients, the boundary element method (BEM) that we adopt is the most appropriate numerical method for discretising the direct and adjoint problems involved in the CGM algorithm. Moreover, all the knowns (input) and unknowns (output) in the inverse problems that we consider are at the boundary, and the discretisation of the boundary only is the essence of the BEM.
The variational problems have been discretised by the BEM and convergence results of the CGM have been proved which add rigour to our investigation. The fellow has served the purpose of transferring knowledge by bringing his unique and solid mathematical background from the outside the EU into the EU. Tested and verified on a wide variety of benchmark examples, the numerically obtained results showed to be accurate, stable and robust, and ready for practical validation.
The research developed in this project will allow to broaden largely the information on the thermal state of structures used and newly designed and will give a possibility to introduce new structures and products, and bring new power-consumable processes to a commercial status.
The fellowship has allowed the fellow to build substantially on his research experience in inverse heat transfer problems and to expose and immerse himself in a vigorous, demanding and active research atmosphere in which collaboration between scientists and engineers is the norm.
The mathematical knowledge that the fellow has brought into Europe on the theoretical analysis of the inversely formulated heat transfer coefficient identification problems has helped making Europe more competitive. The open issues that were addressed and solved in this research proposal make a substantial impact and contribute to the increasing level of fundamental research activity being performed in the inverse theory and numerics in Europe.
The fellowship has facilitated a lasting relationship between the researchers at Leeds and the fellow that will enable the knowledge transfer process to continue beyond the life of the fellowship. The success of this research provides significant additional momentum to develop the theme of determining heat transfer coefficients into a broader investigation encompassing the inverse mathematical modelling of inverse problems and its related computational implementation. As a continuation of this project which both the scientist-in-charge and the fellow envisage, future work will consist in extending some of the inverse analysis performed in this project to composite heat conductors involving the determination of the thermal contact conductance during metal casting, of exhaust gases, and in plate finned-tube heat exchangers.