The aim of intensity modulated radiation therapy (IMRT) is to kill tumor cells while at the same time protecting the surrounding tissue and organs from the damaging effect of radiation. To achieve these goals computerized inverse planning systems are used. Given the number of beams and beam directions, beam intensity profiles that yield the best dose distribution under consideration of clinical and physical constraints are calculated. This is called beam intensity optimization problem. In this thesis, we first review existing mathematical models and computation methods for the beam intensity optimization problem. Next, we formulate the beam intensity optimization problem as a multiobjective linear programme (MOLP) with three objectives. For clinical cases this optimization problem involves thousands of variables and tens of thousands of constraints and existing methods such as multiobjective simplex methods can not handle it. The rest of the thesis is dedicated to developing methods to solve this large MOLP efficiently and to the application in the beam intensity optimization problem. Benson (1998c) argues that solving an MOLP in objective space needs less computation time than solving it in decision space if the number of objectives of the MOLP is much smaller than the number of variables. Moreover, the constraint matrix of the problem relies on the calculation of dose deposited in tissue. Since this calculation is always imprecise solving the MOLP exactly is not necessary in practice. This motivates us to develop algorithms for solving an MOLP in objective space approximately. We summarize Benson’s outer approximation algorithm for solving MOLPs in objective space and propose some small changes to improve computational performance. Then in order to approximate the true nondominated set we propose a modification of Benson’s algorithm which is called an approximation version of Benson’s algorithm. Our approximation algorithm computes an inner and an outer approximation of the nondominated set. We prove that the inner approximation provides a set of ε-nondominated points. The geometric duality theory of Heyde and L¨ohne (2006) defines a dual to an MOLP and it assures us to be able to find the nondominated set of the primal MOLP by solving its dual MOLP. Based on this we develop a dual variant of Benson’s outer approximation algorithm to solve the dual MOLP in objective space. We prove that solving the dual provides a weight set decomposition. We compare the primal algorithm and the dual algorithm on small illustrative and on radiotherapy examples. Furthermore, we propose an algorithm to solve the dual MOLP approximately but within specified tolerance. This approximate solution set can be used to calculate an approximation of the nondominated set of the primal MOLP.We show that this set is an ε-nondominated set of the original primal MOLP and provide numerical evidence that this approach can be faster than solving the primal MOLP approximately. Considering that the set of nondominated points is infinite, it is not very useful from the planners’ point of view. We address the problem of finding well distributed nondominated points for an MOLP.We propose a method which combines the global shooting and normal boundary intersection methods. By doing so, we overcome the limitation of normal boundary intersection method that parts of the nondominated set may be missed. Discrepancy analysis of the nondominated points from a geometry point of view shows that this method produces evenly distributed nondominated points. Moreover, the coverage error and the uniformity level can be measured. Finally, we apply the algorithms developed to the beam intensity optimization problem of 3D clinical cases with voxel size of 5mm and 3mm. A technique of reducing the resolution in normal tissue has been used to reduce the computation time. The results clearly illustrate the advantages of our methods.

One of the most important process operations in the pulp and paper industry is the transport of pulp in pipe lines. Because pipe friction losses are much higher than with water under comparable conditions, accurate design correlations for each pulp are important to the industry. The purpose of this investigation was to design and build a flow rig suitable for investigating a wide range of pulp conditions, to obtain pipe friction loss data for New Zealand pulps, and to produce design correlations and procedures for the industry. This thesis is therefore concerned primarily with describing the experimental equipment and procedures, presenting pipe friction loss data for a variety of New Zealand pulps, including a design correlation for them, and developing design methods for computing friction losses. It includes, in addition, data on drag reduction observed at high velocities of flow, and a discussion of flow mechanisms in each regime of flow. The equipment was designed to produce friction loss data from three pipe diameters simultaneously for each consistency of pulp. Flow rate was controlled without throttling the flow. Pipe friction loss data are presented for five Kraft pulps and one neutral sulphite semi-chemical pulp. Data were obtained from 1,2,3 and 4 in. diameter PVC pipes for a wide range of consistencies and flow rates up to 0.8 ft3/sec. Standard Lampen mill evaluations on hand sheets made from the pulps are presented, as well as data on the characteristics of the fibres. The Kraft pulps exhibited the characteristic maxima and minima but the semi-chemical pulp did not exhibit these turning points. For Kraft pulps head losses before the respective maxima were increased by refining the pulp and using rough pipe; and decreased by adding short-fibre Tawa and by drying and reslushing the pulp. In comparison with maxima for the unbeaten Kraft Pulp, the maxima of the head loss curves for all Kraft pulps were shifted to lower velocities by the above-mentioned operations. This would reduce the friction loss in many practical cases. In particular, rough pipe lowers the magnitude of friction loss in this regime, and can therefore yield a considerable economic advantage. A single design correlation for Kraft pulps is presented for the regime of flow before the maxima in the head loss curves. The limits of the correlation are given. Friction losses of New Zealand pulps were found to be lower than those previously reported in the literature. Two methods of design are presented for the regimes at velocities above the maxima in the head loss curves. A procedure is suggested for pulp and paper mills to obtain their own limits for the design correlation and to verify the correlation proposed in this investigation for their own pulps. A design correlation for the Tawa NSSC pulp is also presented. Mechanisms of flow are discussed for Kraft pulps and a semi-chemical pulp. Visual observations in an artificially roughened pipe for the regime of flow before the maxima of the head loss curves have confirmed fibre-wall contact in this regime. Data obtained at the first sign of permanent plug disruption have been correlated with data at the onset of drag reduction. Fully developed turbulence was found to occur at the maximum level of drag reduction. Some velocity profiles are reported for the transition regime using a modified annular-purge probe. In addition the disruptive shear stress of fibre networks has been correlated by three different methods. Data for the onset of drag reduction are presented and compared with data previously obtained from large diameter pipes from other investigations. This correlation is used as a method for designing piping systems at high flow rates.

This thesis is a study of the large scale flows of water and heat which give rise to the geothermal fields in the Taupo Volcanic Zone (TVZ), New Zealand. To carry out this study, a super-critical equation of state module has been developed for the geothermal simulator TOUGH2, which can describe the flow of water at the conditions expected deep in the TVZ. The code is used to simulate the behaviour of a range of idealised TVZ models in 2D and 3D settings. Hydrothermal plumes which remain stable for periods comparable to the lifetime of the TVZ can occur when there is a contrast between the high permeability of the inner TVZ 'infill' region and the lower permeability exterior region. In this case, downflows of cool surface fluid in the inner TVZ 'sweep' the geothermal heat across the TVZ at depth to the permeability barrier, where the heated fluid ascends to the surface in discrete plumes. This behaviour occurs in 2D models, where separate plumes form at each side of the high permeability infill region, and also in 3D models of caldera-like structures, where perhaps four hot plumes can form around the perimeter of the caldera. This notion is then applied to the complete TVZ hydrological system, where a permeable ‘envelope’ is defined by the location of the Taupo Fault Belt and the currently known volcanic centres in the TVZ. The permeability within this envelope varies spatially according to the geothermal heat flux, and the region outside has relatively low permeability. The spatial variation of the geothermal heat flux is obtained by summing the measured heat flows from the geothermal fields for a number of areas across the TVZ. In this model, the geothermal fields form about the boundary of the envelope, as in the TVZ, and bear a striking resemblance to the actual TVZ geothermal fields. Finally, a new simulation code, NaCl-TOUGH2, is developed to provide a tool for future modelling involving the commonest chemical species in the TVZ - salt. The code incorporates the complete phase diagram for salt-water mixtures and involves liquid, vapour and solid phases over a wide range of temperatures, pressures and salt concentrations. The code is used to solve a number of simple geothermal and mathematical problems.

The forest harvesting problem, FHP, is described. A review of the existing literature is presented along with an analysis of the strengths and limitations of various attempted solutions. The diversity of model evident in recent papers is noted. The difference is explained between a strategic model that sets long-term harvesting goals in terms of total area to be cut each year, and a tactical model that produces a short-term schedule of actual blocks. Special attention is devoted to the development of FRI’s Forestry-Oriented Linear Programming Interpreter, FOLPI, which is currently used to formulate an LP model of the strategic planning problem. Reasons are presented for the desirability of an integrated model, embracing both strategic and tactical decisions, which is capable of optimisation. Accordingly the project then proceeds to a thoughtful and detailed construction of such a model. Particular care is taken to examine the status and function of FOLPI within this model. A column generation algorithm is then developed to solve the relaxed linear program formulation. Finally powerful constraint branching techniques are utilised to obtain the desired optimal solution to the integrated model. Throughout the development of the project the Whangapoua forest in Coromandel, New Zealand has been used as a case study. A concluding section presents numerical output from some of the exhaustive computational analysis associated with this application.

Alumina performs several functions in a modern smelter - it is used to scrub the pot gases to remove fluoride, acts as a thermal insulator on top of the cell and, when added to cryolite-based electrolytes, it is the raw material used to produce aluminium. Alumina is also expected to have good flow and handling properties, and dissolve well in the bath. Unfortunately, it does not always dissolve rapidly and this leads to the formation of "sludge", which creates operating disturbances in the cell; it is difficult to remedy this problem without a basic understanding of the process of alumina dissolution. Consequently, the objective of the work presented in this thesis was to develop an apparatus and technique that would allow the dissolution behaviour of powder alumina to be measured as realistically and objectively as possible, and then determine the important factors affecting dissolution. The method developed involved the merging of three different techniques: ● electroanalytical measurement of dissolved oxide concentrations ● recording of the associated thermal phenomena ● visual observation of the interaction of alumina with the bath which were then used to investigate the effect of a range of alumina properties and smelter operating conditions on dissolution behaviour. A series of dissolution parameters was selected to evaluate the relative dissolution behaviour of the different variables. It was found that slow dissolution behaviour resulted primarily from poor feeding and/or dispersion, coupled with poor heat transfer in the first few seconds of the dissolution process. If the flow properties of the alumina were good, alumina flowed easily out of the dropper on to the surface of the bath forming relatively thin but well-distributed rafts. If not, alumina could flow out into a localized area producing very dense rafts which clumped together and eventually sank to the bottom forming sludge. properties such as loss on ignition enhanced the dispersion of the alumina through the release of volatiles, which caused the alumina to "effervesce" on the surface. Conversely, bulk density aggravated the clumping problem as the density of the formed rafts increased with increasing bulk density. Good feeder design can help to counteract deficiencies in the flow properties of an alumina, if the importance of wide coverage and the imparting of a horizontal velocity component to the alumina are kept in mind. It was also found that if the bath agitation was increased, either by increasing the amount of existing stirring or by reducing the bath volume in the feeding area for a given bath velocity, the initial heat transfer to the alumina could be dramatically improved. The presence of bath superheat was important for maintaining optimum heat transfer conditions for fast dissolutions, as alumina was found to dissolve better when a higher proportion of the heat required was supplied from the bulk of the liquid bath as opposed to localized freezing. Increasing the initial alumina concentration in the electrolyte retarded the dissolution process, with the retardation becoming increasingly more significant at higher concentration values. Similarly, reducing the cryolite ratio, which also decreased the alumina solubility, resulted in the dissolution being inhibited in the later stages, as more alumina dissolved. In situations where the mass transfer in the bath was improved, the impact of these concentration effects was minimized. The video recordings were useful for indicating potential operating difficulties with the samples, such as excessive emissions and flow problems. Six different raft formations, characterizing the degree of bath surface coverage and the raft density or cohesion, were identified from the videos and the aluminas were classified accordingly. Preliminary heat balance calculations were performed using the data obtained in this study and were used to estimate the heat of dissolution for α-Al2O3, for a variety of alumina concentrations in cryolite-based electrolytes. The results indicated heat values of 112±15 and 55±5 kJ mol-1 for alumina concentrations in the bath of 0.43 and 2.83 wt% respectively. Further work is necessary to refine these calculations, however. Reacted ore was found to dissolve more slowly than parent virgin alumina but it was difficult to ascertain which property caused the difference - whether bulk density, flowability, influence of HF, or presence of other impurities. As a result, it is recommended that a series of dissolution runs be performed on reacted samples of different compositions to establish which property associated with reacted ore causes the problem.

The onset of convection in a horizontal fluid layer, heated from below, is, examined by means of perturbation analysis. The resulting eigenvalue system of equations is solved by means of a new extension of a Fourier series technique. Two sets of coupled effects are investigated: (i) thermal buoyancy and surface-tension effects, and (ii) thermal buoyancy and solute buoyancy effects. For the first set of effects the magnetohydrodynamic problem is also studied. For the surface-tension problem, attention is focussed on the case where the lower boundary is a rigid conductor and the upper free surface is subject to a general thermal condition. It is found that for this care the surface-tension and buoyancy forces reinforce each other and are tightly coupled. Cells formed by surface tension are approximately the same size as those formed by buoyancy. The stream line patterns produced by the two agencies acting separately are again similar. When the fluid is electrically conducting and is in the presence of a vertical magnetic field, it is found that the field always has a stabilizing effect. When convection cells are formed in the presence of such a field, their horizontal dimensions are less than for cells formed in the absence of the field. The magnetic field accentuates the difference between the cells induced by surface tension and those by buoyancy, and thus reduces the coupling between the destabilizing forces. Increase of magnetic field causes the buoyancy cell pattern to become more symmetrical, but causes the streamlines in surface-tension cells to become bunched near the surface. When the magnetic field is large, the transition from one type of cell to the other type is extremely sudden, at least when the upper surface is a good thermal conductor. It has been found that, on the model considered, there can be no oscillatory for this problem. However, dimensional analysis reveals that, for a sufficiently flexible upper surface, oscillatory Instability might in fact occur. Finally the thermohaline problem, where the density varies with both temperature and the concentration of some solute, is studied. The eigenvalue equation is now found for general boundary conditions. The degree of coupling between the thermal and the solute effects again depends on the similarity between convection cells caused by the two agencies acting separately. (For one extreme case studied the coupling is zero for a certain range of parameters.) In this problem both monotonic and oscillatory instability can now occur.

The work described in this thesis represents an investigation undertaken with the two-fold purpose of (a) establishing the apparatus and techniques necessary for carrying out investigations into the response of metals to dynamic loading, and (b) examining the response of beta-brass single crystals to dynamic loading. The first part of the investigation required the development of basic apparatus for investigations of the type planned. This apparatus included a dynamic compression testing machine and apparatus for the production and preparation of metal single crystals. The second part of the investigation required the assessing of the existing knowledge of the static and dynamic properties of beta-brass and the planning of a programme aimed at contributing to this knowledge. A review of the existing literature is presented and a programme of tests is outlined. As a result of the investigation the following conclusions were drawn concerning the static and dynamic properties of beta-brass single crystals. The dynamic upper and lower yield stresses were found to be strain-rate sensitive and the upper yield stress increased by up to 4.5 times the static yield stress. The dynamic upper and lower yield stresses were found to be temperature dependent and to show the same temperature dependence as the static yield stress. Both increasing strain-rate and decreasing temperature raised the yield stress. The dynamically deformed material showed large yield drops whereas the statically deformed material did not. Both the static and dynamic work hardening behaviour were found to be orientation dependent. The rate of work hardening for the static tests increased with decreasing temperature whereas that for the dynamic tests did not. The dynamic work hardening behaviour was insensitive to strain-rate. At strains greater than 4 per cent the dynamically deformed material was found to be softer than that statically deformed the same amount. Both the statically and dynamically tested material deformed by slip and the formation of deformation bands. Twinning was not observed. Jerky flow was found to be composition and temperature sensitive.

Aluminium smelting is an energy intensive process, and as a result there has been considerable and ongoing research over a number of decades on the energy efficiency of various aspects of the process. One of the most important measures is current efficiency, which has been shown to have direct relationships with current density, cell temperature, electrolyte chemistry, and anode-cathode distance. The effects of these variables on current efficiency are generally accepted, however there remains debate over the influence of the alumina concentration in the electrolyte on current efficiency. This research relied upon the development of a laboratory scale aluminium smelting cell where the current efficiency was measured via sampling of the product gases. A modified oxygen balance was used, with gas analysis performed using online mass spectrometry. The findings of this research agreed with the accepted current efficiency trends, showing a current density influence of 17.25 %CE per A/cm2, over the range 0.3 and 1.1 A/cm2. The influence of electrolyte chemistry was -7.8 %CE per unit cryolite molar ratio, between cryolite ratios 1.99 and 3. The anode-cathode distance was shown to have no influence on current efficiency in this cell, contradicting the established findings, however this was expected because of the design of the cell with no metal pad at the cathode and therefore constant mass transfer conditions at all the anode-cathode distances used. The most significant finding concerning current efficiency is that the variation with alumina concentration is so small, -0.0376 %CE/wt% Al2O3, that there is effectively no influence. While in many other studies an influence was found, the values and direction of the relationship varied. This suggests that in many cases the observed variation in current efficiency was actually caused by a change in the level of stability in the cell, by processes such as dissolution of sludge from the cathode or the thermal disturbance of alumina feeding, whereas in this research the cell was stable under all operating conditions.

This thesis reports investigations into the stiffness of and energy loss in two series of prestressed concrete beams subjected to reversing sinusoidal cyclic end rotations of equal magnitude and direction. These rotations produced equal moments at the ends of the beams and zero moment at the centre which led to uniform shear along the length of the member. Special care was taken in the manufacture of the two series of beams to ensure essential uniformity of the properties of the beams one to another. The end rotations and the resulting moments where measured by suitable transducers and traces of the moment versus rotation relationship were recorded on an X-Y recorder at intervals during each test run. The amplitude of rotation was progressively changed during each test to ensure a steady value of the end moments. From these recordings the changes in end stiffness and energy absorption were observed and recorded against the number of cycles of moment applied. Concrete strains near the ends of the beams were recorded during the tests. The end moments were expressed as a percentage of the “ultimate” moment which would be required to fail the beams in one reverse cycle of loading. The moment rotation properties of the beams were observed to change as the number of cycles of loading were increased, this change being more rapid for higher percentages of “ultimate” moment. The energy dissipation was found to increase with an increase in the frequency of the cyclic loading, It was considered that the greater part of the energy dissipation arose from the material damping within the concrete. At higher percentages of ultimate moment, the number of cycles sustained before failure was found to increase when ties were inserted around the reinforcing wires. The results from the tests were analysed and an attempt made to relate the properties by the beans to the properties of the component materials of the prestressed members. A theoretical model was postulated with certain simplifications of the relationships of the material properties and a computer program designed to simulate qualitatively, the behaviour of the beams tested. A non-linear hysteretic stress-strain relationship was assumed for the concrete in this analysis which gave similar stress-strain traces to these found in the experimental analysis.

A new model is presented to explain the results of tracer tests of reservoirs. The flow field model is two-dimensional and can include multiple wells, background flow and a high permeability fracture of finite length or a barrier of finite length - likely features of a reservoir in fractured rock. The flow field is represented by a complex potential. This enables the accurate calculation of the time for tracer to travel along the streamline linking the release position and the observation well. The shape of the concentration curve measured at the observation well is governed by dispersion. Two types of dispersion are considered – small scale dispersion and large scale dispersion. Small scale dispersion is due to interaction with the rock matrix and to small scale variations in velocity (not included in the flow field model), in the neighbourhood of a streamline. Large scale dispersion is due to the spread of tracer over streamlines which have different travel times to the observation well and may go to different observation wells. In the literature, small scale dispersion models are used to match the concentration curves for individual wells. Here it is shown that models with differing small scale mechanisms give equally good matches to Wairakei data and that response curves for small scale dispersion and large scale dispersion are similar. This means that the shape of a concentration response curve gives little information about reservoir structure. The important results from a tracer test are whether tracer is detected at each observation well and the travel times to the wells that respond, in conjunction with the locations of the release and production wells. This new approach to analysing tracer tests is used together with the flow field model to deduce the permeability structure and flow field of a simple hypothetical reservoir and of the Wairakei (New Zealand) geothermal reservoir.