Yes / It is shown that the probability generating function of a lifetime random variable T on a finite lattice with polynomial failure rate satisfies a certain differential equation. The interrelationship with Markov chain theory is highlighted. The differential equation gives rise to a system of differential equations which, when inverted, can be used in the limit to express the polynomial coefficients in terms of the factorial moments of T. This then can be used to estimate the polynomial coefficients. Some special cases are worked through symbolically using Computer Algebra. A simulation study is used to validate the approach and to explore its potential in the reliability context.

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En 2010 Chung, Graham et Knuth ont démontré une remarquable identité symétrique sur les nombres eulériens et posé le problème de trouver un q-analogue de leur identité. En utilisant les q-polynômes eulériens introduits par Shareshian-Wachs, nous avons pu obtenir une telle q-identité. La preuve bijective que nous avons imaginée, nous a permis ensuite de démontrer d'autres q-identités symétriques, en utilisant un modèle combinatoire dû à Foata-Han. Entre temps, Hyatt a introduit les fonctions quasisymétriques eulériennes colorées afin d'étudier la distribution conjointe du nombre d'excédances et de l'indice majeur sur les permutations colorées. En appliquant le Decrease Value Theorem de Foata-Han, nous donnons d'abord une nouvelle preuve de sa formule principale sur la fonction génératrice des fonctions quasisymétriques eulériennes colorées, puis généralisons certaines identités eulériennes symétriques, en les exprimant comme des identités sur les fonctions quasisymétriques eulériennes colorées. D'autre part, en prolongeant les travaux récents de Savage-Visontai et Bec-raun, nous considérons plusieurs q-polynômes de descente des mots signés. Leurs fonctions génératrices factorielles et multivariées sont explicitement calculées. Par ailleurs, nous montrons que certains de ces polynômes n'ont que des zéros réels. Enfin, nous étudions la fonction génératrice diagonale des nombres de Jacobi Stirling de deuxième espèce, en généralisant des résultats analogues pour les nombres de Stirling et Legendre-Stirling de deuxième espèce. Il s'avère que cette fonction génératrice est une série rationnelle dont le numérateur est un polynôme à coefficients entiers positifs. En appliquant la théorie des P-partitions de Stanley nous trouvons des interprétations combinatoires de ces coefficients / In 2010 Chung-Graham-Knuth proved an interesting symmetric identity for the Eulerian numbers and asked for a q-analog version. Using the q-Eulerian polynomials introduced by Shareshian-Wachs we find such a q-identity. Moreover, we provide a bijective proof that we further generalize to prove other symmetric qidentities using a combinatorial model due to Foata-Han. Meanwhile, Hyatt has introduced the colored Eulerian quasisymmetric functions to study the joint distribution of the excedance number and major index on colored permutations. Using the Decrease Value Theorem of Foata-Han we give a new proof of his main generating function formula for the colored Eulerian quasisymmetric functions. Furthermore, certain symmetric q-Eulerian identities are generalized and expressed as identities involving the colored Eulerian quasisymmetric functions. Next, generalizing the recent works of Savage-Visontai and Beck-Braun we investigate some q-descent polynomials of general signed multipermutations. The factorial and multivariate generating functions for these q-descent polynomials are obtained and the real rootedness results of some of these polynomials are given. Finally, we study the diagonal generating function of the Jacobi-Stirling numbers of the second kind by generalizing the analogous results for the Stirling and Legendre-Stirling numbers of the second kind. It turns out that the generating function is a rational function, whose numerator is a polynomial with nonnegative integral coefficients. By applying Stanley’s theory of P-partitions we find combinatorial interpretations of those coefficients

La micro cogénération désigne la génération simultanée de deux types d’énergie à faible puissance. En énergétique, ce terme désigne en pratique la production simultanée d’électricité et de chaleur : le principe reposant sur la récupération de la chaleur fatale induite par la production électrique. Deux bancs d’essais ont d’abord été réalisés sur deux prototypes de micro cogénérateurs : un moteur Stirling à gaz et un moteur à vapeur à granulés de bois. Une campagne expérimentale a été menée pour caractériser chaque système au niveau énergétique et environnemental. Les résultats expérimentaux ont abouti sur deux modèles numériques dynamiques et semi-physiques de micro cogénérateurs programmés dans l’environnement numérique TRNSYS où une plateforme numérique de simulation a été développée. Celle-ci intègre principalement des modèles de systèmes de stockage d’énergie, des générateurs stochastiques de fichiers de besoins énergétiques et des stratégies innovantes de pilotage des systèmes et des charges selon des critères de précision et de réalisme.Cette plateforme a permis d’évaluer la pertinence énergétique, environnementale et économique de micro cogénérateurs couplés aux bâtiments d’habitation et au réseau électrique selon différentes configurations. / Micro combined heat and power (µCHP) or cogeneration means the simultaneous generation of two energy types. In energetic fields, this term refers usually to the simultaneous production of electricity and heat: the principle being based on the recovering of the fatal heat induced by the electricity production processes.Firstly, two test benches were carried out on two µCHP prototypes: a gas Stirling engine and a wood pellets steam engine. Experimental investigations were conducted to characterize each system at energy and environmental levels. The experimental results led two dynamic and semi physical numerical models of µCHP systems programmed in the numerical tool TRNSYS where a numerical platform has been developed. This platform integrates mainly energy storage systems models, stochastic energy needs file generators and innovative management strategy of systems and energy loads according to precision and realism criteria.This platform allows assessing realistic energy, environmental and economic relevance of µCHP systems coupled with dwelling buildings and the power grid according to different configurations.

Plusieurs machines actuellement utilisées, moteurs à combustion interne en automobile ou centrales thermiques dans l’énergie, rejettent de grandes quantités de chaleur. Généralement cette chaleur est dissipée dans l’atmosphère et son énergie perdue. Nous nous sommes donc intéressés aux moteurs à apport de chaleur externe dont l’énergie primaire est de l’énergie thermique, et plus particulièrement aux moteurs Stirling. L’une de ses principales caractéristiques est d’utiliser de la chaleur produite extérieurement comme source d’énergie. Ceci lui permet d’être multi-carburant et même d’utiliser de l’énergie thermique naturelle.L’étude menée comporte deux parties. Tout d’abord un modèle numérique zéro dimension, trois zones en temps fini a été développé. Il prend en compte les échanges thermiques aux parois et les pertes de charge, mais ne préjuge ni des dimensions moteur, ni des conditions de fonctionnement. Ceci lui permet de rester flexible pour s’adapter à l’architecture spécifique du moteur à simuler. Ensuite nous avons réalisé des mesures expérimentales sur deux moteurs de taille et puissance différentes (quelques watts et 1 kW). Ces résultats ont permis de valider le modèle. Au final nous avons obtenu un modèle numérique traduisant l’influence de paramètres dimensionnels et fonctionnels sur la puissance du moteur Stirling.Un outil d’aide à la conception de moteur Stirling a été développé en ajoutant au modèle un algorithme d’optimisation. Il permet une ébauche des caractéristiques d’un moteur Stirling. En fonction de l’application souhaitée et des contraintes s’y appliquant, il agit sur les caractéristiques choisies par l’utilisateur pour maximiser les performances. / Several machines currently used, internal combustion engines for the car industry or thermal power plants in energy, exhaust a considerable amount of heat. Generally this heat is dispersed in the atmosphere and its energy lost. So we took a special interest in external heat engines which primary energy is heat energy, and more particularly in Stirling engines. One of its main characteristics is the used of energy from heat produced externally like energy source. This allows Stirling engines to be multi-fuel and even to use natural heat energy.The study carried out is made up of two parts. First, a three zones zero dimensional finite-time thermodynamic model has been developed. It takes into account the heat transfer from the walls and the pressure drop, but does not prejudge the dimensions of the engine nor the conditions of its functioning. It is thus able to remain flexible and to adjust to the specific architecture of the engine that should be simulated. Afterwards, we have realized a series of experimental measures thanks to two engines different in size and power (a few Watt and 1 kW). These results allowed us to validate the model. In the end, we got a numerical model representing the influence of dimensional and functional parameters on the power of a Stirling engine.Eventually, a tool to assist in designing Stirling engines was developed adding an optimization algorithm to the model. It allows to sketch out a preliminary draft of the characteristics of a Stirling engine. Depending on the desired application and on the constraints exerted on the engine, the tool created will act on the characteristics of the engine chosen by the user to maximize its performances.

Thesis (MScEng)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: Several transverse flux and longitudinal flux linear generator topologies exist for freepiston Stirling engine applications. In this thesis the transverse flux permanent magnet linear generators are investigated together with a back-to-back converter which can deliver the electrical energy from the linear generator to the electrical network. The transverse flux permanent magnet linear generator is geometrically optimised with the aim to maximise the power-to-weight ratio while maintaining preset power and efficiency levels. An optimised 3 kW linear generator is built and the measured results correlate to the simulation results. A close-loop current control scheme is introduced to control the current of the rectifier, which is part of the back to back converter. The transverse flux permanent magnet linear generator is connected to the input of the rectifier which has the ability to force a specific current from the generator. The measured results of the rectifier correlate to the results of the simulations that were done. The current control present some complications and it is suggested that another control scheme is used. A close-loop voltage control scheme is introduced for the control of the DC bus voltage. The DC bus is connected between the rectifier and the inverter, which is the other part of the back-to-back converter. A close-loop current control scheme is introduced to control the inverter current that flows from the inverter to the electrical network. The measured results of the inverter and the DC bus correlate to the results of the simulations that were done. The results of the system, including the generator, rectifier and inverter, tested as a unit is presented and discussed. / AFRIKAANSE OPSOMMING: Verskeie tranverse vloed en longitudinale vloed lineˆere generator topologie¨e bestaan vir vrysuier Stirling enjin toepassings. In hierdie tesis word ’n transverse vloed permanente magneet lineˆere generator ondersoek saam met ’n omsetter. Die omsetter dra die elektriese energie van die generator oor aan die elektriese netwerk. Die transverse vloed permanente magneet lineˆere generator word geometries geoptimeer met die doel om die drywing-tot-gewig verhouding te maksimiseer terwyl vasgestelde drywing en effektiwiteit vlakke behou word. ’n Geoptimeerde 3kW lineˆere generator prototipe is vervaardig en die gemete resultate is geverifieer met die simulasie resultate. ’n Geslote lus stroombeheer strategie word voorgestel om die stroom te beheer van die gelykrigter, wat deel is van die omsetter. Die transverse vloed permanente magneet lineˆere generator word aan die gelykrigter, wat die vermo¨e het om ’n spesifieke stroom uit die generator te forseer, se intree verbind. Die gemete resultate van die gelykrigter wat gebou is stem goed ooreen met die van die simulasies wat gedoen is. Die stroombeheer hou komplikasies in wat bespreek word. Dus word die gebruik van ’n alternatiewe stroombeheer voorgestel. ’n Geslote lus spannings beheer strategie¨e word voorgestel om die gs. busspanning te beheer. Die gs. bus is gekonnekteer tussen die gelykrigter en die wisselrigter, wat ook deel uitmaak van die omsetter. ’n Geslote lus stroom beheer word voorgestel om die stroom te beheer wat vanaf die wisselrigter na die elektriese netwerk toe vloei. Die gemete resultate van die wisselrigter en die gs. bus stem goed ooreen met die van die simulasies wat gedoen is. Die resultate van die hele stelsel, wat die generator, gelykrigter en die wisselrigter insluit, wat as ’n eenheid getoets is word weergegee en bespreek.

Large amounts of energy from heat sources such as waste-eat and geothermal energy are available worldwide but their potential for useful power-generation is largely untapped. This is because they are relatively low temperature difference (LTD) sources, in the range from 100 to 200 °C, and it is thermodynamically diffcult, for theoretical and practical reasons, to extract useful work at these temperatures. This work explores the suitability of a Stirling engine (SE) to exploit these heat sources. Elsewhere much work has been done to optimise Stirling engines for high temperature heat sources, but little is known about suitable engine layouts, and their optimal design and operational aspects at lower temperature differences. With the reduced temperature difference, changes from conventional engine designs become necessary and robust solutions for this novel application have to be identified. This has been achieved in four major steps: identification of a suitable engine type; thermodynamic optimisation of operating and engine parameters; optimisation of mechanical efficiency; and the development of conceptual designs for the engine and its components informed by the preceding analysis. For the optimisation of engine and operating parameters a model was set up in the commercial Stirling software package, Sage, which also has been validated in this thesis; suitable parameter combinations have been identified. This work makes key contributions in several areas. This first is the identification of methods for better simulating the thermodynamic behaviour of these engines. At low temperature differences the performance of Stirling engines is very sensitive to losses by fluid friction (and thus frequency), adiabatic temperature rise during compression, and the heat transfer from and to the surroundings. Consequently the usual isothermal analytical approaches produce results that can be misleading. It is necessary to use a non-isothermal approach, and the work shows how this may be achieved. A second contribution is the identification of the important design variables and their causal effects on system performance. The primary design variable is engine layout. For an engine having inherently low efficiency due to the low temperature difference it is important to choose the engine layout that provides the highest power density possible in order to minimise engine size and to save costs. From this analysis the double-acting alpha-type configuration has been identified as being the most suitable, as opposed to the beta or gamma configurations. An-other key design variable is working fluid, and the results identify helium and hydrogen as suitable, and air and nitrogen as unsuitable. Frequency and phase angle are other design variables, and the work identifies favourable values. A sensitivity analysis identifies the phase angle, regenerator porosity, and temperature levels as the most sensitive parameters for power and efficiency. It has also been shown that the compression work in low-temperature difference Stirling engines is of similar magnitude as the expansion work. By compounding suitable working spaces on one piston the net forces on the piston rod can be reduced significantly. In double-acting alpha-engines this can be achieved by choosing the Siemens as opposed to the Franchot arrangement. As a result friction and piston seal leakage which are two important loss mechanisms are reduced significantly and longevity and mechanical efficiency is enhanced. Design implications are identified for various components, including pistons, seals, heat exchangers, regenerator, power extraction, and crankcase. The peculiarities of the heat source are also taken into account in these design recommendations. A third key contribution is the extraction of novel insights from the modelling process. For the heat exchangers it has been shown that the hot and cold heat exchangers can be identical in their design without any negative impact on performance for the low-temperature difference situation. In comparison the high temperature applications invariably require different materials and designs for the two heat exchangers. Also, frequency and phase angle are found to be quite different (lower frequency and higher phase angle) from the optimum parameters found in high temperature engines. Contrary to common belief the role of dead volume has been found to play a crucial and not necessary detrimental role at low temperature differentials. Taken together, the work is positioned at the intersection of thermodynamic analysis and engineering design, for the challenging area of Stirling engines at low temperature differences. The work extracts thermodynamic insights and extends these into design implications. Together these help create a robust theoretical and design foundation for further research and development in the important area of energy recovery.