In this dissertation we discuss three problems. We first show the classical q-Stirling numbers of the second kind can be expressed more compactly as a pair of statistics on a subset of restricted growth words. We extend this enumerative result via a decomposition of a new poset which we call the Stirling poset of the second kind. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the q-Stirling numbers of the first kind is done. We also give a bijective argument showing the (q, t)-Stirling numbers of the first and second kind are orthogonal. In the second part we give combinatorial proofs of q-Stirling identities via restricted growth words. This includes new proofs of the generating function of q-Stirling numbers of the second kind, the q-Vandermonde convolution for Stirling numbers and the q-Frobenius identity. A poset theoretic proof of Carlitz’s identity is also included. In the last part we discuss a new expression for q-binomial coefficients based on the weighting of certain 01-permutations via a new bistatistic related to the major index. We also show that the bistatistics between the inversion number and major index are equidistributed. We generalize this idea to q-multinomial coefficients evaluated at negative q values. An instance of the cyclic sieving phenomenon related to flags of unitary spaces is also studied.

Thesis (MEng) -- Stellenbosch University, 2014. / ENGLISH ABSTRACT: The aim of this study was to design and manufacture an experimentally testable free-piston Stirling engine (FPSE), including a linear electric generator; to develop and validate a theoretical simulation model; to identify problem areas pertaining to its manufacture; and finally to assess the work undertaken, to lay out the groundwork for the future development of a 3 kWe FPSE suitable for incorporation in a solar Stirling dish power generator. A redesigned version of the Beale B- 10B demonstrator engine was manufactured to overcome design diffculties and to simplify testing. The design made use of an electric generator designed at the Department of Electrical and Electronic Engineering at Stellenbosch University. Experimental measurements included piston and displacer motions, hot side and cold side temperatures, working space pressure, electric generator output, as well as heat rejection via a water jacket. Experimental measurements were taken prior to and subsequent to the addition of the electric generator. Indicated power was calculated as 0,659 W at a frequency of 10,99 Hz prior to the addition of the electric generator. The addition of the electric generator was unsuccessful since it was not well matched with the engine. The indicated power calculated was between 0,138 W and 0,144 W for different loads on the electric generator, while the electrical output power ranged from 1,23 mWe to 1,79 mWe. The addition of the electric generator produced non-continuous motion caused by magnetic forces instead of engine pressure variations. The major manufacturing diffculty was the attachment of magnets for the electric generator, but this was overcome with the manufacture of a special assembly jig. The theoretical simulation model was a combination of a third-order and dynamic analysis. Working space values were solved by the application of the conservation of mass, momentum and energy equations for a one-dimensional discretised model of the engine, while the motion of the piston and displacer was determined by applying the equations of motion. The majority of experimental measurements were predicted more accurately when higher heat transfer coeficients were used between the working space and wall temperatures. The theoretical simulation model was used to gain insight into the effect of input parameters on engine operation. The displacer rod diameter was shown to have implications on output power and stability, while it was shown that there is a natural tendency to deliver constant output power at a near-constant frequency over a range of piston loads for an FPSE. It was also shown that the design of an FPSE is complex and that the design of all components should be done in parallel. The control of an FPSE was seen to be both a necessity and can be used to exploit the advantages of the uncoupled nature of an FPSE. / AFRIKKANSE OPSOMMING: Die doel van hierdie studie was om 'n eksperimentele toetsbare vrye-werksuier Stirling enjin te vervaardiging, wat 'n lineêre elektriese kragopwekker insluit; om 'n teoretiese simulasie model te ontwikkel en te yk; om vervaardiging probleme te identi seer; en om die ondernemende werk te assesseer om 'n fondasie te lê vir die toekomstige ontwikkeling van 'n 3 kWe vrye-werksuier Stirling enjin wat by 'n Stirling sonskottel ingelyf kan word. 'n Herontwerpte weergawe van die Beale B-10B demonstrasie enjin was vervaardig om ontwerp probleme te bowe te kom en om die toets daarvan te vereenvoudig. Die ontwerp het gebruik gemaak van 'n elektriese kragopwekker wat by die Departement Elektriese en Elektroniese Ingenieurswese aan die Universiteit van Stellenbosch ontwerp is. Eksperimentele metings het die werksuier en verplaser bewegings ingesluit, sowel as die warm kant en koue kant temperature, die werkruimte druk, die elektriese uitset van die kragopwekker, sowel as die hitteuitruiling wat met 'n water verkoelingskringloop gepaard gaan. Eksperimentele metings was geneem voor en na die byvoeging van die elektriese kragopwekker. Kraglewering was bereken op 0,659 W teen 'n frekwensie van 10,99 Hz voordat die elektriese kragopwekker bygevoeg is. Die byvoeging van die elektriese kragopwekker was onsuksesvol omdat die nie gepas was vir die enjin nie. Die kraglewering is bereken op vlakke wat gewissel het tussen 0,138 W en 0,144 W vir die verskillende belastings op die elektriese kragopwekker, terwyl die elektriese uitset gewissel het tussen 1,23 mWe en 1,79 mWe. Die byvoeging van die elektriese kragopwekker het 'n nie-aaneenlopende beweging veroorsaak weens die magnetiese kragte wat dit beinvloed het in plaas van enjindruk variasies. Die belangrikste ontwerpuitdagings was die ontwerp van 'n werksuier en verplaser wat 'n klein toleransie passing kon handhaaf om sodoende 'n seël te verseker terwyl dit aan temperatuur variasies blootgestel was. Die grootste vervaardigingsprobleem was die aanheg van magnete vir die elektriese kragopwekker, maar dit is te bowe gekom deur 'n spesiale voeg te vervaardig. Die teoretiese simulasie model was 'n kombinasie van 'n derde-orde en 'n dinamiese analise. Werkruimte waardes was opgelos deur die toepassing van die behoud van massa, momentum en energie vergelykings vir 'n een-dimensionele gediskretiseerde model van die enjin, terwyl die beweging van die werksuier en verplaser bepaal was deur die toepassing van die bewegingvergelykings. Die meerderheid van die eksperimentele metings was meer akkuraat voorspel wanneer hoër warmteoordrag koë siënte tussen die werkruimte en muurtemperature gebruik was. Die teoretiese simulasie model was gebruik om insig in terme van die e ek van invoer veranderlikes op die enjin gedrag te toon. Daar was getoon dat die verplaserstaaf diameter implikasies het op kragoplewering en stabiliteit, terwyl die natuurlike tendens van 'n vrye-werksuier Stirling enjin gewys was om 'n konstante kraguitvoer te lewer op 'n naby-konstante frekwensie oor 'n reeks werksuier laste. Daar was ook gewys dat die ontwerp van 'n vryewerksuier Stirling enjin kompleks is en dat die ontwerp van alle komponente in parallel gedoen moet word. Die beheer van 'n vrye-werksuier Stirling enjin was gewys om beide noodsaaklik te wees, sowel as gebruik kan word om die unieke voordele van 'n vrye-werksuier Stirling enjin se ongekoppelde natuur te ontgin.

This dissertation is divided into four chapters. In Chapter 2 the equivariant homology groups of upper order ideals in the partition lattice are computed. The homology groups of these filters are written in terms of border strip Specht modules as well as in terms of links in an associated complex in the lattice of compositions. The classification is used to reproduce topological calculations of many well-studied subcomplexes of the partition lattice, including the d-divisible partition lattice and the Frobenius complex. In Chapter 3 the box polynomial B_{m,n}(x) is defined in terms of all integer partitions that fit in an m by n box. The real roots of the box polynomial are completely characterized, and an asymptotically tight bound on the norms of the complex roots is also given. An equivalent definition of the box polynomial is given via applications of the finite difference operator Delta to the monomial x^{m+n}. The box polynomials are also used to find identities counting set partitions with all even or odd blocks, respectively. Chapter 4 extends results from Chapter 3 to give combinatorial proofs for the ordinary generating function for set partitions with all even or all odd block sizes, respectively. This is achieved by looking at a multivariable generating function analog of the Stirling numbers of the second kind using restricted growth words. Chapter 5 introduces a colored variant of the ordered partition lattice, denoted Q_n^{\alpha}, as well an associated complex known as the alpha-colored permutahedron, whose face poset is Q_n^\alpha. Connections between the Eulerian polynomials and Stirling numbers of the second kind are developed via the fibers of a map from Q_n^{\alpha} to the symmetric group on n-elements

A computational model for a resonantly coupled alpha free-piston Stirling cooler is presented. The cooler consists of two isothermal working spaces for compression and expansion connected by a regenerator consisting of a stack of narrow parallel channels. The regenerator is assumed to have a linear temperature distribution along its axial direction and the working fluid is taken as an ideal gas. Control volume analysis is adapted in this model, in which each of the components of the cooler is considered a separate control volume. The compression piston is given a predetermined motion to provide the work needed by the cooler. The expansion piston and the gas trapped between the piston and the walls of the expansion cylinder are modeled as a mass, spring, and damper system. The motion of the compression piston generates a pressure difference across the cooler, and forces the working fluid to pass through the regenerator. The expansion piston responds to the pressure in its space according to Newton's second law of motion. The motion of the expansion piston is governed by the forces originating from the pressure and the cold side gas spring and dash-pot. In this way the dynamics of the moving pistons are coupled to the thermodynamics of the cooler system. A definition for the coefficient of performance (COP) that considers the heat transfer by conduction through the material making up the regenerator is introduced. This definition of the COP reflects the dependence of the cooler's performance on the length of the regenerator. From a systematic variation of this regenerator length, an optimal value can be found for a given set of operating parameters. Conservation laws of mass, momentum and energy along with ideal gas relations are used to form a set of equations fully describing the motion of the pistons and the thermal state of the cooler. A marching-in-time technique with a Runge-Kutta scheme of the fourth order is adapted to integrate the equation of motion. The plots of the motion of the pistons, the pressure-volume diagrams of the workspaces and the COP plots are provided to describe the cooler behavior. / Graduation date: 1999

Thermoacoustic engines convert heat into acoustic pressure waves with no moving parts; this inherently results in high reliability, low maintenance and low manufacturing costs. Significant increases in the performance of these devices have enabled rivalry with more mature energy conversion methods in both efficiency and power output. This optimal production of acoustic power can be ultimately used to achieve cryogenic temperatures in thermoacoustic refrigerators, or can be interfaced with reciprocating electro-acoustic power transducers to generate electricity. This thesis describes the design, fabrication and testing of a Thermoacoustic Power Converter. The system interfaces a thermoacoustic-Stirling heat engine with a pair of linear alternators to produce 100 watts of electricity from a heat input. It operates with helium at 450 psig internal pressure and a hot side temperature of 1200F. Through thermoacoustic phenomena, these conditions sustain a powerful pressure wave at a system specific 100 Hz. This pressure wave is used to drive the two opposed linear alternators in equal and opposite directions to produce a single phase AC electrical output at that same system frequency. The opposing motion of the two alternators enables a vibration-balanced system. The engine has created 110 watts of acoustic power and the complete Thermoacoustic Power Converter system has produced 70 watts of AC electricity. Compensating for some heat leaks, the converter reaches 26.3% heat to acoustic power efficiency and 16.8% heat to electric efficiency when those maximum values are achieved. This conversion of heat to acoustic power is 40% of the Carnot thermodynamic efficiency limit.

Stirling cryocoolers are required for a wide variety of applications, especially in military equipment, due to their small size, low weight, long lifetime and high reliability considering their efficiency. Thus, it is important to be able to investigate the operating performance of these coolers in the design stage. This study focuses on developing a computer program for simulating a Stirling cryocooler according to the second order analysis. The main consideration is to simulate thermodynamic, fluid dynamic and heat transfer behavior of Stirling cryocoolers. This goal is achieved by following the route of Urieli (1984), which was focused on Stirling cycle engines. In this research, a simulation for performance prediction of a Stirling cryocooler is performed. In addition to that, the effects of system parameters are investigated. This attempt helps to understand the real behavior of Stirling cryocoolers using porous regenerator material. Results implied that first order analysis methods give optimistic predictions where second order method provides more realistic data compared to first order methods. In addition to that, it is shown that regenerator porosity has positive effect on heat transfer characteristics while affecting flow friction negatively. As a conclusion, this study provides a clear understanding of loss mechanisms in a cryocooler. Performed numerical analysis can be used as a tool for investigation of effects of system parameters on overall performance.

Stirling cycle cryocoolers are used widely in military applications. The regenerator is the key element of Stirling cycle cryocoolers. It is known that performance of the regenerator directly affects the cryocooler performance. Therefore, any improvement on the regenerator will lead to a more efficient cryocooler. Thus, it is essential to have an idea about regenerator parameters and their effects on the system. In this study Stirling engine regenerator, which is constructed by wire mesh screens, is accepted as a porous medium. Using energy balance and continuity equation, matrix and fluid thermal equations are derived. Simplified versions of these equations are obtained for not only the ideal case, but also two other cases which take into account the effects of longitudinal conduction and the effects of regenerator wall. A computer code is developed in Matlab to solve these equations using finite difference method. The developed code is validated by using Sage. Afterwards, effects of all regenerator parameters on regenerator performance are investigated in detail and results are presented. To make this investigation easier, a graphical user interface is also built (in Matlab) and used.

Runtime enforcement mechanisms are an important and well-employed method for ensuring an execution only exhibits acceptable behavior, as dictated by a security policy. Wherever interaction occurs between two or more parties that do not completely trust each other, it is most often the case that a runtime enforcement mechanism is between them in some form, monitoring the exchange. Considering the ubiquity of such scenarios in the computing world, there has been an increased effort to build formal models of runtime monitors that closely capture their capabilities so that their effectiveness can be analysed more precisely. While models have grown more faithful to their real-life counterparts, is- sues concerning complexity and manageability (a common concern for software engineers) of centralized policies remains to be fully addressed. The goal of this thesis is to provide a principled approach to policy construction that is modular, intuitive, and backed by formal methods. This thesis introduces a class of policy combinators adequate for use with runtime en- forcement policies and analyses a particular instance of them called Static Committee Com- binators (SCCs). SCCs present a model of policy composition where combinators act as committees that vote on events passing through the monitor. They were conceptualized in collaboration with Jay Ligatti and Daniel Lomsak. The general class of combinators are called Static Decision Combinators (SDCs), which share key features with SCCs such as allowing combinators to respond with alternative events when polled, in addition to re- sponding with grants or denials. SDCs treat the base-level policies they compose as black boxes, which helps decouple the system of combinators from the underlying policy model. The base policies could be modelled by automata but the combinators would not maintain their own state, being "static". This allows them to be easily defined and understood using truth tables, as well as analysed using logic tools. In addition to an analysis of SDCs and SCCs, we provide useful examples and a reusable combinator library.

There are many sources of free energy available in the form of heat that is often simply wasted for want of an effective way to convert it into useful energy such as electricity. The aim of this research project is to design and build a low temperature differential Stirling engine capable of generating electric power from heat sources such as waste hot water or geothermal springs. The engine that has been developed is a research prototype model of a new type of design featuring a rotating displacer which is actuated by a pair of stepper motors. The rotating displacer design enables the use of readily available and comparatively cheap and robust steam pipe as the housing for the engine, and it also avoids problems associated with sealing and heat exchange that would be present in a large engine of a more traditional configuration. Owing to the fact that this engine is a research prototype, it has the ability to have some of its critical operating parameters such as phase angle and stroke length adjusted to investigate the effects on performance. When the next phase of development takes place most of these parameters will be fixed at the optimum values which will make manufacture cheaper and easier. Unfortunately, construction of the prototype engine has not been completed at the time of writing so no power producing results have been achieved; however thorough results are presented on the operation of the control system for the stepper motors which actuate the displacer. Additionally, after a thorough history and background of Stirling engines was researched, the understanding gained of how these engines work has enabled a design process to take place which has hopefully led to a successful design. Analysis of various aspects of the engine have been carried out and results look promising for the engine to produce around 500 Watts of electrical power output whilst running on hot water up to around 90°C.

Chip scale refrigeration system is critical for the development of electronics with the rapid increase of power consumption and substantial reduction of device size, resulting in an emergent demand on novel cooling technologies with a high efficiency for the thermal management. In this thesis, active refrigeration devices based on Stirling cycle and an electrocaloric material, are designed and investigated to achieve a high cooling performance. Firstly, a new Stirling micro-refrigeration system composed of arrays of silicon MEMS cooling elements is designed and evaluated. The cooling elements are fabricated in a stacked array on a silicon wafer. A regenerator is placed between the compression (hot side) and expansion (cold side) diaphragms, which are driven electrostatically. Under operating conditions, the hot and cold diaphragms oscillate sinusoidally and out of phase such that heat is extracted to the expansion space and released from the compression space. A first-order of thermodynamic analysis is performed to study the effect of geometric parameters. Losses due to regenerator non-idealities and chamber heat transfer limitation are estimated. A multiphysics computational approach for analyzing the system performance that considers compressible flow and heat transfer with a large deformable mesh is demonstrated. The optimal regenerator porosity for the best system COP (coefficient of performance) is identified. To overcome the computational complexity brought about by the fine pillar structure in the regenerator, a porous medium model is used to allow for modeling of a full element. The analysis indicates the work recovery of the system and the diaphragm actuation are main challenges for this cooler design.The pressure drop and friction factor of gas flow across circular silicon micro pillar arrays fabricated by deep reactive ion etch (DRIE) process are investigated. A new correlation that considers the coupled effect of pillar spacing and aspect ratio, is proposed to predict the friction factor in a Reynolds v number range of 1-100. Silicon pillars with large artificial roughness amplitudes is also fabricated, and the effect of the roughness is studied in the laminar flow region. The significant reduction of pressure drop and friction factor indicates that a large artificial roughness could be built for pillar arrays in the regenerator to enhance the micro-cooler efficiency. The second option is to develop a fluid-based refrigeration system using an electrocaloric material poly(vinylidene fluoride-trifluoroethylene-chlorofluoroethylene) [P(VDF-TrFE-CFE)] terpolymer. Each cooling element includes two diaphragm actuators fabricated in the plane of a silicon wafer, which drive a heat transfer fluid back and forth across terpolymer layers that are placed between them. Finite element simulations with an assumption of sinusoidal diaphrahm motions are conducted to explore the system performance detailedly, including the effects of the applied electric field, geometric dimensions, operating frequency and externally-applied temperature span. Multiphysics modeling coupled with solid-fluid interaction, heat transfer, electrostatics, porous medium and moving mesh technique is successfully performed to verify the thermal modeling feasibility. The electrocaloric effect in thin films of P(VDF-TrFE-CFE) terpolymer is directly measured by infrared imaging at ambient conditions. At an electric field of 90 V/μm, an adiabatic temperature change of 5.2 °C is obtained and the material performance is stable over a long testing period. These results suggest that application of this terpolymer is promising for micro-scale refrigeration.