Thermal weakening friction during seismic slip experiments and models with heat sources and sinks

Experiments that systematically explore rock friction under crustal earthquake conditions reveal that faults undergo abrupt dynamic weakening. Processes related to heating and weakening of fault surfaces have been invoked to explain pronounced velocity weakening. Both contact asperity temperature Ta and background temperature T of the slip zone evolve significantly during high-velocity slip due to heat sources (frictional work), heat sinks (e.g., latent heat of decomposition processes), and diffusion. Using carefully calibrated High-Velocity Rotary Friction experiments, we test the compatibility of thermal weakening models: (1) a model of friction based only on T in an extremely simplified, Arrhenius-like thermal dependence; (2) a flash heating model which accounts for the evolution of both V and T; (3) same but including heat sinks in the thermal balance; and (4) same but including the thermal dependence of diffusivity and heat capacity. All models reflect the experimental results but model (1) results in unrealistically low temperatures and model (2) reproduces the restrengthening phase only by modifying the parameters for each experimental condition. The presence of dissipative heat sinks in stage (3) significantly affects T and reflects on the friction, allowing a better joint fit of the initial weakening and final strength recovery across a range of experiments. Temperature is significantly altered by thermal dependence of (4). However, similar results can be obtained by (3) and (4) by adjusting the energy sinks. To compute temperature in this type of problem, we compare the efficiency of three different numerical approximations (finite difference, wavenumber summation, and discrete integral)

Optimal control of semiconductor melts by traveling magnetic fields

In this paper, the optimal control of traveling magnetic fields in a process of crystal growth from the melt of semiconductor materials is considered. As controls, the phase shifts of the voltage in the coils of a heater-magnet module are employed to generate Lorentz forces for stirring the crystal melt in an optimal way. By the use of a new industrial heater-magnet module, the Lorentz forces have a stronger impact on the melt than in earlier technologies. It is known from experiments that during the growth process temperature oscillations with respect to time occur in the neighborhood of the solid-liquid interface. These oscillations may strongly influence the quality of the growing single crystal. As it seems to be impossible to suppress them completely, the main goal of optimization has to be less ambitious, namely, one tries to achieve oscillations that have a small amplitude and a frequency which is sufficiently high such that the solid-liquid interface does not have enough time to react to the oscillations. In our approach, we control the oscillations at a finite number of selected points in the neighborhood of the solidification front. The system dynamics is modeled by a coupled system of partial differential equations that account for instationary heat condution, turbulent melt flow, and magnetic field. We report on numerical methods for solving this system and for the optimization of the whole process. Different objective functionals are tested to reach the goal of optimization

Optimal control of semiconductor melts by traveling magnetic fields

In this paper, the optimal control of traveling magnetic fields in a process of crystal growth from the melt of semiconductor materials is considered. As controls, the phase shifts of the voltage in the coils of a heater-magnet module are employed to generate Lorentz forces for stirring the crystal melt in an optimal way. By the use of a new industrial heater-magnet module, the Lorentz forces have a stronger impact on the melt than in earlier technologies. It is known from experiments that during the growth process temperature oscillations with respect to time occur in the neighborhood of the solid-liquid interface. These oscillations may strongly influence the quality of the growing single crystal. As it seems to be impossible to suppress them completely, the main goal of optimization has to be less ambitious, namely, one tries to achieve oscillations that have a small amplitude and a frequency which is sufficiently high such that the solid-liquid interface does not have enough time to react to the oscillations. In our approach, we control the oscillations at a finite number of selected points in the neighborhood of the solidification front. The system dynamics is modeled by a coupled system of partial differential equations that account for instationary heat condution, turbulent melt flow, and magnetic field. We report on numerical methods for solving this system and for the optimization of the whole process. Different objective functionals are tested to reach the goal of optimization

Numerical Computation of Moving Boundary Phenomena

When matter is subjected to a gradient of: temperature, pressure, concentration, voltage or chemical potential a phase change may occur, which for dynamic processes will be separated by moving boundaries between the adjacent phases. Transport properties vary considerably between phases, consequently any change in phase modifies the rate of transport of: energy, momentum, charge or matter which are fundamental to the behaviour of many physical systems. Such dynamic multi-phase problems have, for historical and mathematical reasons, become known as either: Stefan problems or Moving Boundary Problems (MBPs). In most engineering applications the analysis of these problems is often impossible without recourse to numerical schemes which utilise either: finite difference or finite element methods. The success of finite element methods is their ability to handle complex geometries; however, they are time consuming and less amenable to vectorisation than finite difference techniques which, because of their greater simplicity in formulation and programming, continue to be the more popular choice. Several finite difference schemes are available for the solution of moving boundary problems; however, there are some difficulties associated with each method. Each time a new numerical scheme is developed, it has the aim of improving either, or both, the accuracy and the computational performance. For solving one-dimensional moving boundary problems, the variable time step grid is the best approach in terms of simplicity and computational efficiency. Due to the fact that the time step is variable the implicit recurrence formulae, which are stable for any mesh size, have always been used with this type of discretisation of the space time domain. It will be shown in the course of this thesis that the implicit methods are very inaccurate when used with relatively large time steps; hence, the immediate conclusion may be made - that implicit variable time step methods may not be sufficiently accurate to solve moving boundary problems where the boundary is moving with a relatively slow velocity. The proposed idea, of combining real and virtual grid networks and using new explicit finite difference equations, eliminates the loss of accuracy associated with implicit methods, when the time step is large, and offers higher computational performance. The new finite difference equations are based on the approach of making the finite difference substitution into the solution of the partial differential equation rather than into the partial differential equation itself, which is the classical approach. A new numerical scheme for two-phase Stefan problems which will be referred to as the EVTS method is developed and the solution is compared to other numerical methods as well as the analytic solution. Furthermore, the EVTS method is modified to solve implicit moving boundary problems (oxygen diffusion problem), in which an explicit relation containing the velocity of the moving boundary is absent. The resulting method achieves similar results to other more complex and time consuming methods. A further numerical scheme referred to as the ZC method is developed to deal with heat transfer problems involving three phases (or 2 moving boundaries) which appear and disappear during the process. To the knowledge of the author, a finite difference method for such a problem does not exist. For validation, numerical results are compared with those of the conservative finite element method of Bonnerot and Jamet, which is the only other method available to solve two-moving boundary problems. Finally, a new finite difference solution for non-linear problems is developed and applied to laser heat treatment of materials. The numerical results are in good agreement with published experimental results

Large-Eddy Simulations of Flow and Heat Transfer in Complex Three-Dimensional Multilouvered Fins

The paper describes the computational procedure and results from large-eddy simulations in a complex three-dimensional louver geometry. The three-dimensionality in the louver geometry occurs along the height of the fin, where the angled louver transitions to the flat landing and joins with the tube surface. The transition region is characterized by a swept leading edge and decreasing flow area between louvers. Preliminary results show a high energy compact vortex jet forming in this region. The jet forms in the vicinity of the louver junction with the flat landing and is drawn under the louver in the transition region. Its interaction with the surface of the louver produces vorticity of the opposite sign, which aids in augmenting heat transfer on the louver surface. The top surface of the louver in the transition region experiences large velocities in the vicinity of the surface and exhibits higher heat transfer coefficients than the bottom surface.Air Conditioning and Refrigeration Project 9

Mathematical modelling of fixed bed reactors

Consideration is given to the solution of the highly exothermic fixed bed catalytic reactor problem taking into account heat and mass transfer resistances inside the catalyst pellets and across the external fluid film as well as radial temperature and oonoentration gradients in the fluid phase. Comparison of the model with the simpler quasi homogeneous repreaenation is made. In the region where the quasi homogeneous case predicts temperature "run-away", the added refinements assume some importance. Very significant; differences in behaviour are predicted. Indeed no temperature "run-away" is apparent. Inolucling simply a film mass and heat transfer resistance is no guarantee that temperature "run-away" will not be predicted. In fact, it is the particle diffusive resistance whioh is the main factor limiting the temperature effects. Since the region of temperature "run-away" is often in the practical range it is essential to use a more detailed model for design such as the one described here, especially if optimal operating conditions are being sought. Even on large digital computers, the computation time is excessively long if the sets of differential equations are solved simultaneously. By examining the intrapartiole equations in detail for a practical range of physical properties and operating conditions, it is shown that they may be reduced, to a lumped parameter form. While still retaining the characteristics of the general problem, the lumped parameter approximation can be solved in a substantially shorter time, thus taking its use in optimization and control studies feasible

Non-isothermal dynamics of thin-film free-surface and channel flows of non-Newtonian nanofluids

Numerical modelling of the dynamic behaviour of generalized-viscoelastic-fluidbased nanofluids (GVFBNs) and viscoelastic-fluid-based nanofluids (VFBNs) has a number of industrial applications such as in new battery technologies and phasechange heat transfer devices. The computational results have shown that for certain flow parameters values, some of the non-Newtonian fluids also known as complex fluids (e.g. worm-like micellar solutions, granular flows, polymer solutions and some polymer melts) reveal flow instabilities within the flow field, such as the emergence of regions of different shear bands due to the flow induced material non-homogeneities. It has also been observed that it is becoming increasingly clear that the thermal runway phenomenon should not be ignored in polymers or other complex fluids since it may, in some instances, be as important as the complex rheology in differentiating susceptibility order for different types of nanofluids, for instance Newtonian fluid Based Nanofluids (NFBN), Generalized Newtonian Fluid-Based Nanofluids (GNFBN), Viscoelastic-fluid based nanofluids (VFBN) and Generalized viscoelastic fluid based nanofluids (GVFBN). These computational observations laid the foundation of this thesis. We have investigated the improvement of heat transfer for GVFBN and VFBN by homogenously mixed spherical shape nanoparticles. To incorporate the nanoparticles in the governing equations we use a single phase nanofluid modelling approach. Our mathematical models are governed by a system of non-linear, highly coupled, time-dependent Partial Differential Equations (PDEs). We developed computational solutions in Matlab software for the resulting system of equations by using an efficient semi-implicit finite-difference method, combined with a Crank-Nicolson scheme. In addition, the effects of nanoparticles on fluid velocity, extra stresses, temperature, and thermal conductivity are explored. Comparisons of the numerical results for the nanofluids with those from the literature without nanoparticles show excellent agreement

Modelling analysis of heat transfer in polymeric materials exposed to different heating scenarios

Polymers undergo physical, chemical and structural changes when exposed to heat and/or fire. Thermoplastics melt, decompose and burn; thermosets decompose, char and/or burn, depending on the temperature changes due to external incident heat flux. Detailed in this thesis is a theoretical and numerical heat transfer study, which is undertaken to simulate and experimentally validated temperature variations during melting, decomposition, charring and ignition phases of polymers. For melting, thermoplastic polymers (polypropylene, polyester, polyamide 6, polymethyl methacrylate, polycarbonate and polystyrene) have been used, whereas for decomposition, charring and ignition glass fibre – reinforced epoxy composites have been chosen. For each case a one-dimensional finite difference method, using Matlab as the operator has been developed to determine the transient temperature distributions within the different types of polymers materials. The convective and radiative heat transfer boundary conditions, at the exposed and unexposed sides of polymer samples, have also been taken into account accordingly. While some experimental results to validate the different numerical models built are from other researchers’ work at Bolton, in addition to these, other sets of experiments were specifically developed for this work. The melting behaviour of thermoplastics has been modelled in two scenarios: (i) vertically oriented sample where melt dripping occurs and (ii) horizontally oriented sample within a contained holder in order that the mass will not escape from the containment region. In the the first scenario the sample was placed in a tube furnace, where the radiant heat is uniform on all sides of the sample. This is based on the experimental methodology developed at Bolton University in an earlier project which studied the melt dripping behaviour of polymers. The thermogravimetric and rheological analysis of molten drops had indicated that, depending upon the temperature of the furnace (external heat flux) and the structure of the polymer, in some cases it was pure melting whereas in others it was accompanied by a partial decomposition of the polymer. A one-dimensional finite difference method based on a moving boundary approach has been developed to model the temperatures of the molten drops polymers. The simulated results showed good agreement with the molten drops’ temperatures measured by experiments. In addition, using kinetic parameters, degrees of decomposition in drops obtained at different furnace temperatures were also simulated, which were validated with previous experimental results. For the second scenario, in which the sample is placed horizontally in a container, experiments were conducted using a cone calorimeter with the heat applied only on the top surface, while the other sides of the polymer sample are insulated, A further one-dimensional finite difference method based on a Stefan approach involving phase changing material, has been developed to determine the melting point temperature and to estimate the temperature profile within the polymer slab, to simulate pure melting and melting plus partial decomposition which may or may not catch fire depending upon the degree of decomposition. The predicted results matched well with the experimental results. Furthermore, the heat transfer model was modified to simulate the temperature profiles through the thickness of a glass fibre - reinforced composite exposed to different heat fluxes in a cone calorimeter. This involved incorporating a kinetic model for the decomposition process taking into consideration the varying thermophysical properties as a function of temperature. This is achieved by using the critical heat flux that is the minimum incident heat flux leading to ignition, in the equation defining the ignition temperature, The simulated temperature profiles matched well with the experimental results obtained from previous works at the University of Bolton, giving a much better agreement than previously published models describing this condition. Ignition phenomenon is well described by the model showing a jumping step when the composite polymer ignites and burns. The last part of the work was to simulate the heat transfer in Intumescent coated glass fibre reinforced epoxy composites exposed to heat in a cone calorimeter. On exposure to heat the intumescent coating expands to form a char, the thickness and the thermal conductivity of which, depends on the type of coating. It was not the purpose of this work to model expansion of the coating; rather the emphasis was to understand the thermal barrier efficiency of the expanded char. However, changes to the surface, expansion of the local thickness and char region when exposed to heat were incorporated into the model to gain better agreement with experiment values

Design and development of a new time integration framework, GS4-1, and its application to silica particle deposition

Growing interest in the simulation of first order transient systems, typical of those encountered in transient heat conduction, flow transport, and fluid dynamics, has prompted the development of a variety of time integration methods for solving these systems numerically. The primary contribution of this thesis is the design and development of a new time integration/discretization framework, under the class of single step single solve algorithms which are the most popular, for use in such first order transient systems with computationally attractive features. These include second order accuracy, unconditional stability, zero-order overshoot, and controllable numerical dissipation with a new selective control feature which overcomes the restrictions in the existing and current state-of-the-art methods. Throughout the thesis, we demonstrate the capability and advantage of the newly developed framework, termed GS4-1, in comparison to existing methods using various types of numerical examples (both linear and nonlinear). The numerical results consistently demonstrate the roles played by the new feature in improving the numerical solutions of both the primary variable and its time derivative which is important to correctly capture the dynamics of the problems, in contrast to the existing methods without such a feature. Additionally, a breakthrough contribution presented in this thesis is the development of an isochronous integration framework (iIntegrator), stemming from the novel relations between the newly developed GS4-1 framework and the existing GS4-2 framework (for second order dynamic systems). Such a development enables the use of the same computational framework to solve both first and second order dynamic systems without having to resort to the individual GS4-1 and GS4-2 frameworks; hence the practicality in the computational and implementation aspects. Finally, the application of the new GS4-1 framework to silica particle deposition, which is a practical problem of interest, is presented with the focus primarily on the physics of the problem. In this part of the thesis, a numerical model of the problem is presented and employed to investigate the effects of the flow and physicochemical parameters on the rate of deposition. The results of the parametric studies undertaken based on the employed numerical model enable some recommendations for the mitigation of the problem, and therefore serve as additional valuable contribution of the thesis