Faculty of Engineering | Te Kaupeka Pūhanga
https://hdl.handle.net/10092/1
Thu, 29 Sep 2022 11:51:32 GMT2022-09-29T11:51:32ZThe first three chapters of the Grahalāghava of Gaṇeśa Daivajña : edition, translation, and mathematical and historical analysis.
https://hdl.handle.net/10092/104500
The first three chapters of the Grahalāghava of Gaṇeśa Daivajña : edition, translation, and mathematical and historical analysis.
Cidambi, Sahana
The field of history of mathematics allows mathematicians the ability to trace patterns of mathematical development
throughout human history, and often reconnect them to the inspirations and motivations that led
to mathematical progress. The narrative of mathematical development in India, specifically
the second millennium Indian exact sciences, has only been a topic of close study in recent years. This classical era of mathematical astronomy in India featured a diverse expanse of calculatory works
devoted to predictive astronomy, or finding the timings, locations, and appearances of celestial phenomena.
The choice of parameters for time divisions and number of cyclic planetary revolutions for these calculations
primarily differentiated the different schools during this era.
One such school, the Gaṇeśapakṣa, was founded by Gaṇeśa Daivajña (b. 1507 CE) of Nandigrāma, India
with the composition of his text, the Grahalāghava (“brevity [in calculations] of the planets”) (1520 CE).
The text featured innovative modifications to the previously established time divisions and mean longitude
calculations, as well as the ambitious removal of trigonometric computations in all stated formulas. These
parameters and procedures inspired a sizable proliferation of commentaries and astronomical tables through
much of the second half of the second millennium, with many extant manuscripts and much relevance to
astrology in India today.
In this thesis, we use critical editions and manuscripts of the Grahalāghava and earlier astronomical
works to study the first three chapters of the Grahalāghava with the aim of exploring Gaṇeśa’s trigonometryapproximating
techniques and tracing potential influences of earlier works to contextualize the Grahalāghava
in the larger tradition.
Sat, 01 Jan 2022 00:00:00 GMThttps://hdl.handle.net/10092/1045002022-01-01T00:00:00ZA novel method to design monolithic catalysts for non-isothermal packed bed reactors.
https://hdl.handle.net/10092/104499
A novel method to design monolithic catalysts for non-isothermal packed bed reactors.
Erfani, Navid
As the popularity of additive manufacturing grows, modern design techniques such as topology optimisation are gaining more attention. Topology optimisation has utilised the freedom offered by additive manufacturing to create complex and non-intuitive shapes for structural design problems. Recent developments have made it possible to use this method to solve conjugate heat transfer problems such as heat sink designs. However, work addressing chemical engineering design problems with topology optimisation is scarce, especially when dealing with problems involving modelling combined heat, mass, and momentum transfer.
This research aims to extend topology optimisation to the design of complex monolithic catalysts for packed bed reactors with endothermic reactions. This study makes an original contribution to the field by including mass transfer limitations alongside the coupled momentum, heat, and mass transfer equations. A new form of the Arrhenius equation tailored to the density-based method is also presented. Additionally, a combination of interpolation functions is proposed and implemented successfully in the optimisation. The present study is the first to report a large 3D topology optimisation of a monolithic catalyst for packed bed reactors, which also validates the optimisation by comparison with a conventional monolithic structure.
First, in the context of a simpler 2D heat transfer topology optimisation problem, two main categories of non-gradient-based and gradient-based methods of optimisation algorithms were explored. A topology optimisation method based on a genetic algorithm was developed and used to solve the heat transfer optimisation problem, and the results were benchmarked against the density-based method. The density-based method relying on a gradient-based optimisation algorithm significantly outperformed the non-gradient-based genetic algorithm in terms of computation time and obtained objective function value.
In the next step, a more complex reactor model was developed using the modified Navier-Stokes equations for modelling the laminar fluid flow and a convection-diffusion equation to model the heat transfer. The experimental data from the literature were used to validate the approach for simulating the conjugate heat transfer in the reactor. Then, a convection-diffusion equation describing the mass transfer and a first-order mass transfer limited endothermic reaction were included in the model. Using the density-based method, the optimisation problem was formulated as a catalyst distribution problem. The catalyst volume fraction of elements in the discretised design domain determined the geometry of the monolithic structure and consequently the momentum, heat, and mass transfer characteristics of the reactor. With the proposed combination of interpolation functions, the continuous material properties and the sufficient penalisation for the intermediate density values were derived. The Globally Convergent Method of Moving Asymptotes (GCMMA) was employed to solve the optimisation problem.
The effects of initial guesses, the energy dissipation constraint, and the Peclet number on the design were investigated in a 2D framework. Then through a 3D optimisation, a complex structure with enhanced mass transfer, thermal, and hydraulic behaviour was developed. Compared to the reference honeycomb geometry with the same catalyst volume fraction, the resulting 3D optimised geometry showed quantitative improvements in terms of the conversion and pressure drop.
Sat, 01 Jan 2022 00:00:00 GMThttps://hdl.handle.net/10092/1044992022-01-01T00:00:00ZIntersection between natural and artificial swimmers: a scaling approach to underwater vehicle design.
https://hdl.handle.net/10092/104494
Intersection between natural and artificial swimmers: a scaling approach to underwater vehicle design.
Coe, Michael Joseph
Approximately 72% of the Earth’s surface is covered by water, yet only 20% has been mapped [1]. Autonomous Underwater Vehicles (AUVs) are one of the main tools for ocean exploration. The demand for AUVs is expected to increase rapidly in the coming years [2], so there is a need for faster and more energy efficient AUVs. A drawback to using this type of vehicle is the finite amount of energy that is stored onboard in the form of batteries. Science and roboticists have been studying nature for ways to move more efficiently. Phillips et al. [3] presents data that contradicts the idea that fish are better swimmers than conventional AUVs when comparing the energetic cost of swimming in the form of the Cost of Transport (COT). The data presented by Phillips et al. only applies to AUVs at higher length and naval displacement (mass) scales, so the question arises of whether an AUV built at different displacements and length scales is more efficient than biological animals and if current bio-inspired platforms are better than conventional AUVs.
Besides power requirements, it is also useful to compare the kinematic parameters of natural and artificial swimmers. In this case, kinematic parameters indicate how fast the swimmer travels through the water. Also, they describe how fast the propulsion mechanism must act to reach a certain swimming speed. This research adopts the approach of Gazzola et al. [4] where the Reynolds number is associated with a dimensionless number, Swim number (Sw) in this case, that has all the kinematic information. A newly developed number that extends the swim number to conventional AUVs is the Propulsion number (Jw), which demonstrates excellent agreement with the kinematics of conventional AUVs. Despite being functionally similar, Sw and Jw do not have a one-to-one relationship. Sw, Jw, COT represent key performance metrics for an AUV, herein called performance criteria, which can be used to compare existing platforms with each other and estimate the performance of non-existent designs.
The scaling laws are derived by evaluating the performance of 229 biological animals, 163 bioinspire platforms, and 109 conventional AUVs. AUVs and bio-inspired platforms have scarce data compared with biological swimmers. Only 5% of conventional and 38% of bio-inspired AUVs have kinematic data while 30% of conventional and 18% of bio-inspired AUVs have energetic data. The low amount of performance criteria data is due to the nature of most conventional AUVs as commercial products. Only recently has the COT metric been included in the performance criteria for bio-inspired AUVs. For this reason, the research here formulates everything in terms of allometric scaling laws. This type of formulation is used extensively when referring to biological systems and is defined by an exponential relationship f (x) = axb, where x is a physical parameter of the fish or vehicle, like length or displacement. Scaling laws have the added benefit of allowing comparisons with limited data, as is the case for AUVs.
The length and displacement scale (physical scale) must be established before estimating the performance criteria. Scale is primarily determined by the payload needed for a particular application. For instance, surveying the water column in deep water will require different scientific tools than taking images of an oyster bed in an estuary. There is no way to identify the size of an AUV until it is designed for that application, since these scientific instruments each have their own volume, length, and weight. A methodology for estimating physical parameters using computer vision is presented to help determine the scale for the vehicle. It allows accurate scaling of physical parameters of biological and bio-inspired swimmers with only a side and top view of the platform. A physical scale can also be determined based on the vehicle’s overall volume, which is useful when determining how much payload is needed for a particular application. Further, this can be used in conjunction with 3D modeling software to scale nonexistent platforms.
Following the establishment of a physical scale, which locomotion mode would be most appropriate? Unlike conventional AUVs that use propeller or glider locomotion, bio-inspired platforms use a variety of modes. Kinematics and energy expenditures are different for each of these modes. For bio-inspired vehicles, the focus will be on the body-caudal fin (BCF) locomotion, of which four types exist: anguilliform, carangiform, thunniform, and ostraciiform. There is ample research on anguilliform and carangiform locomotion modes, but little research on thunniform and ostraciiform modes. In order to determine which locomotion mode scales best for a bio-inspired AUV, this research examines the power output and kinematic parameters for all four BCF modes. In order to achieve this, computational fluid dynamics simulations are performed on a 2D swimmer for all four modes. Overset meshes are used in lieu of body-fitted meshes to increase stability and decrease computational time. These simulations were used to scale output power over several decades of Reynolds numbers for each locomotion mode. Carangiform locomotion was found to be the most energy efficient, followed by anguilliform, thunniform, and ostraciiform.
In order to utilize the above scaling laws in designing a novel platform, or comparing an existing one, there must be a unifying framework. The framework for choosing a suitable platform is presented with a case study of two bio-inspired vehicles and a conventional one. The framework begins by determining how the platform can be physically scaled depending on the payload. Based on the physical scale and derived scaling laws, it then determines performance criteria. It also describes a method for relative cost scaling for each vehicle, which is not covered in the literature. The cost scaling is based on the assumption that all payloads and materials are the same. The case study shows that a conventional AUV performs better on all performance criteria and would cost less to build.
Sat, 01 Jan 2022 00:00:00 GMThttps://hdl.handle.net/10092/1044942022-01-01T00:00:00ZThe intersection of longest paths in a graph.
https://hdl.handle.net/10092/104484
The intersection of longest paths in a graph.
Mark, Sarah Jayne
In this thesis we examine the famous conjecture that every three longest paths in a graph intersect,
and add to the classes of graphs for which it is known that this conjecture holds. This conjecture arose
from a question asked by Gallai in 1966, the question of whether all of the longest paths in a graph
intersect (Gallai's question). In 1969, Walther found a graph in which the longest paths do not all
intersect, answering Gallai's question. Since then, many other graphs in which the longest paths do not
all intersect have been found. However there are also many classes of graphs for which the longest paths
all intersect, such as series-parallel graphs and dually chordal graphs. Finding such classes of graphs is
an active area of research and in this thesis we add to these classes of graphs.
We begin by investigating Gallai's question for a speci c class of graphs. A theta graph is a graph
consisting of three paths with a pair of common endpoints and no other common vertices. A generalised
theta graph is a graph with at least one block that consists of at least three paths with a pair of common
endpoints and no other common vertices. We show that for a subclass of generalised theta graphs, all of
the longest paths intersect.
Next, we consider the conjecture that every three longest paths of a graph intersect. We prove that,
for every graph with n vertices and at most n + 5 edges, every three longest paths intersect.
Finally, we use computational methods to investigate whether all longest paths intersect, or every
three longest paths intersect, for several classes of graphs. Two graphs are homeomorphic if each can be
obtained from the same graph H by a series of subdivisions. We show that, for every simple connected
graph G that is homeomorphic to a simple connected graph with at most 7 vertices, all of the longest
paths of G intersect. Additionally, we show that, for every simple connected graph G homeomorphic
to a simple connected graph with n vertices, n + 6 edges, and minimum vertex degree 3, all of the
longest paths of G intersect. We then show that for every graph with n vertices and at most n + 5
edges, every three longest paths intersect, independently verifying this result. We also present results
for several additional classes of graphs with conditions on the blocks, maximum degree of the vertices,
and other properties of the graph, showing that every three longest paths intersect or every six longest
paths intersect for these graphs.
Sat, 01 Jan 2022 00:00:00 GMThttps://hdl.handle.net/10092/1044842022-01-01T00:00:00Z