Mathematical aspects of thermoacoustics

This thesis addresses the mathematical aspects of thermoacoustics, a subfield within physical acoustics that comprises all effects in which heat conduction and entropy variations of the gaseous medium play a role. We focus specifically on the theoretical basis of two kinds of devices: the thermoacoustic prime mover, that uses heat to produce sound, and the thermoacoustic heat pump or refrigerator, that use sound to produce heating or cooling. Two kinds of geometry are considered. The first one is the so-called standing-wave geometry that consists of a closed straight tube (the resonator) with a porous medium (the stack) placed inside. The second one is the so-called traveling-wave geometry that consists of a resonator attached to a looped tube with a porous medium (regenerator) placed inside. The stack and the regenerator differ in the sense that the regenerator uses thinner pores to ensure perfect thermal contact. The stack or regenerator can in principle have any arbitrary shape, but are modeled as a collecting of long narrow arbitrarily shaped pores. If the purpose of the device is to generate cooling or heating, then usually a speaker is attached to the regenerator to generate the necessary sound. By means of a systematic approach based on small-parameter asymptotics and dimensional analysis, we have derived a general theory for the thermal and acoustic behavior in a pore. First a linear theory is derived, predicting the thermoacoustic behavior between two closely placed parallel plates. Then the theory is extended by considering arbitrarily shaped pores with the only restriction that the pore cross-sections vary slowly in longitudinal direction. Finally, the theory is completed by the inclusion of nonlinear second-order effects such as streaming, higher harmonics, and shock-waves. It is shown how the presence of any of these nonlinear phenomena (negatively) affects the performance of the device. The final step in the analysis is the linking of the sound field in the stack or regenerator to that of the main tube. For the standing-wave device this is rather straightforward, but for the traveling-wave device all sorts of complications arise due to the complicated geometry. A numerical optimization routine has been developed that chooses the right geometry to ensure that all variables match continuously across every interface and the right flow behavior is attained at the position of the regenerator. Doing so, we can predict the flow behavior throughout the device and validate it against experimental data. The numerical routine can be a valuable aid in the design of traveling-wave devices; by variation of the relevant problem parameters one can look for the optimal travelingwave geometry in terms of power output or efficiency

Weakly nonlinear thermoacoustics for stacks with slowly varying pore cross-sections

A general theory of thermoacoustics is derived for arbitrary stack pores. Previous theoretical treatments of porous media are extended by considering arbitrarily shaped pores with the only restriction that the pore cross-sections vary slowly in the longitudinal direction. No boundary-layer approximation is necessary. Furthermore, the model allows temperature variations in the pore wall. By means of a systematic approach based on dimensional analysis and small parameter asymptotics, we derive a set of ordinary differential equations for the mean temperature and the acoustic pressure and velocity, where the equation for the mean temperature follows as a consistency condition of the assumed asymptotic expansion. The problem of determining the transverse variation is reduced to finding a Green's function for a modified Helmholtz equation and solving two additional integral equations. Similarly the derivation of streaming is reduced to finding a single Green's function for the Poisson equation on the desired geometr

Systematic derivation of the weakly non-linear theory of thermoacoustic devices

Thermoacoustics is the field concerned with transformations between thermal and acoustic energy. This paper teaches the fundamentals of two kinds of thermoacoustic devices: the thermoacoustic prime mover and the thermoacoustic heat pump or refrigerator. Two technologies, involving standing wave and traveling wave modes, are considered. We will investigate the case of a porous medium and two heat exchangers placed in a gas-filled resonator, in which either a standing or traveling wave is maintained. The central problem is the interaction between the porous medium and the sound field in the tube. The conventional thermoacoustic theory is reexamined and a systematic and consistent weakly non-linear theory is constructed based on dimensional analysis and small parameter asymptotics. The difference with conventional thermoacoustic theory lies in the dimensional analysis. This is a powerful tool in understanding physical effects which are coupled to several dimensionless parameters that appear in the equations, such as theMach number, the Prandtl number, the Laucret number and several geometrical quantities. By carefully analyzing limiting situations in which these parameters differ in orders of magnitude, we can study the behavior of the system as a function of parameters connected to geometry, heat transport and viscous effects

Weakly nonlinear thermoacoustics for stacks with slowly varying pore cross-sections

A general theory of thermoacoustics is derived for arbitrary stack pores. Previous theoretical treatments of porous media are extended by considering arbitrarily shaped pores with the only restriction that the pore cross-sections vary slowly in the longitudinal direction. No boundary-layer approximation is necessary. Furthermore, the model allows temperature variations in the pore wall. By means of a systematic approach based on dimensional analysis and small parameter asymptotics, we derive a set of ordinary differential equations for the mean temperature and the acoustic pressure and velocity, where the equation for the mean temperature follows as a consistency condition of the assumed asymptotic expansion. The problem of determining the transverse variation is reduced to finding a Green's function for a modified Helmholtz equation and solving two additional integral equations. Similarly the derivation of streaming is reduced to finding a single Green's function for the Poisson equation on the desired geometry. © Cambridge University Press 2008

Weakly nonlinear thermoacoustics for stacks with slowly varying pore cross-sections

A general theory of thermoacoustics is derived for arbitrary stack pores. Previous theoretical treatments of porous media are extended by considering arbitrarily shaped pores with the only restriction that the pore cross-sections vary slowly in the longitudinal direction. No boundary-layer approximation is necessary. Furthermore, the model allows temperature variations in the pore wall. By means of a systematic approach based on dimensional analysis and small parameter asymptotics, we derive a set of ordinary differential equations for the mean temperature and the acoustic pressure and velocity, where the equation for the mean temperature follows as a consistency condition of the assumed asymptotic expansion. The problem of determining the transverse variation is reduced to finding a Green's function for a modified Helmholtz equation and solving two additional integral equations. Similarly the derivation of streaming is reduced to finding a single Green's function for the Poisson equation on the desired geometry. © Cambridge University Press 2008

Mathematical problems for Moineau pumps

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Mathematical problems for Moineau pumps

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Catching gas with droplets : modelling and simulation of a diffusion-reaction process

The packaging industry wants to produce a foil for food packaging purposes, which is transparent and lets very little oxygen pass. To accomplish this they add a scavenger material to the foil which reacts with the oxygen that diffuses through the foil. We model this process by a system of partial differential equations: a reaction-diffusion equation for the oxygen concentration and a reaction equation for the scavenger concentration. A probabilistic background of this model is given and different methods are used to get information from the model. Homogenization theory is used to describe the influence of the shape of the scavenger droplets on the oxygen flux, an argument using the Fourier number of the foil leads to insight into the dependency on the position of the scavenger and a method via conformal mappings is proposed to find out more about the role of the size of the droplet. Also simulations with Mathematica were done, leading to comparisons between different placements and shapes of the scavenger material in one- and two-dimensional foils

Catching gas with droplets : modelling and simulation of a diffusion-reaction process

The packaging industry wants to produce a foil for food packaging purposes, which is transparent and lets very little oxygen pass. To accomplish this they add a scavenger material to the foil which reacts with the oxygen that diffuses through the foil. We model this process by a system of partial differential equations: a reaction-diffusion equation for the oxygen concentration and a reaction equation for the scavenger concentration. A probabilistic background of this model is given and different methods are used to get information from the model. Homogenization theory is used to describe the influence of the shape of the scavenger droplets on the oxygen flux, an argument using the Fourier number of the foil leads to insight into the dependency on the position of the scavenger and a method via conformal mappings is proposed to find out more about the role of the size of the droplet. Also simulations with Mathematica were done, leading to comparisons between different placements and shapes of the scavenger material in one- and two-dimensional foils