In this dissertation, we study the partial and generic uniqueness of block term tensor decompositions in signal processing. We present several conditions for generic uniqueness of tensor decompositions of multilinear rank (1, L1, L1), ..., (1, LR, LR) terms. Our proof is based on algebraic geometric methods. Mathematical preliminaries for this dissertation are multilinear algebra, and classical algebraic geometry. In geometric language, we prove that the joins of relevant subspace varieties are not tangentially weakly defective. We also give conditions for partial uniqueness of block term tensor decompositions by proving that the joins of relevant subspace varieties are not defective. The main result is the following. For a tensor Y belong to the tensor product of three complex vector spaces of dimensions I, J, K, we assume that L1, L2, ..., LR is from small to large, K is bigger or equal to J, and J is strictly bigger than LR. If the dimension of ambient space is strictly less than IJK, then for general tensors among those admitting block term tensor decomposition, the block term tensor decomposition is partially unique under the condition that the binomial coefficient indexed by J and LR is bigger or equal to R, and I is bigger or equal to 2; it has infinitely many expressions under the condition IJK is strictly less than the sum from L_1^2 to L_R^2; it is essentially unique under any of the following there conditions: (i) I is bigger or equal to 2, J, K is bigger or equal to the sum from L1 to LR (ii) R is 2, I is bigger or equal to 2 (iii) I is bigger or equal to R, K is bigger or equal to the sum from L1 to LR, J is bigger or equal to 2LR, the binomial coefficient indexed by J and LR is bigger or equal to R.

Large scale power generation commences with the combustion of coal or other fuel, which in turn converts high pressure water into steam which then drives a turbine thus generating electricity. Burning high moisture coal, such as lignite, for power generation implies that a significant amount of energy is wasted in vaporising the moisture, which could otherwise be used in the steam raising process. This implies that more moist coal would be required to drive the same process than if the coal was drier, thus increasing the amount of combustion products such as greenhouses gases. Introducing a dried coal in an existing boiler will significantly change the heat flux profiles, which could result in boiler damage or excessive fouling. Flame temperature is influenced by the supply of reactants; in most cases the limiting reactant will be oxygen. The supply of oxygen (through air) to a pneumatically transported coal stream and subsequent reaction is controlled by the localised fluid mechanics or ‘mixing’. This research aims to provide an understanding of the mixing process between the pneumatically transported coal and air in brown coal fired boilers by modelling the individual jets. The effects of the change in velocity ratio for the air (secondary) jets and fuel (primary) jets of rectangular burners typical of those found in brown coal fired boilers has been studied experimentally and is reported in this thesis. In particular, scientific analysis was used to investigate the physical mechanisms which control fuel-air mixing, and to quantify the concentration of primary and secondary fluid. The concentration data was used in a regression model in conjunction with a reactive combustion model, developed from a 1:30 scale cold model of the Yallourn W’ stage 2 boiler, in order that overall boiler performance can be assessed. This overall study is fundamental as a result of the questions raised concerning the future of brown coal in modern society. A qualitative flow visualisation study of the unconfined 1:30 scaled primary, and two adjacent rectangular jets, was conducted using single colour planar laser induced fluorescence. The characteristics of the jet flow were examined by imaging individually seeded primary and secondary jets and were visualised through four different planes longitudinally, on the axes of each jet. In addition, a transverse qualitative and quantitative study on the rectangular jets was also conducted for the individually seeded jets, and was visualised through planes of flow perpendicular to the direction flow, specifically at axial stations of x/D =0.1, 0.2, 0.5, 1, 2, 4, 6 and 8. The flow characteristics were also examined under different co-flow conditions, particularly secondary to primary jet velocity ratios (λ) of 0, 0.55, 1.4, 2.8, 3.6 and ∞. This quantitative data yields the basis for a 3D regression model to predict fuel-air mixing in actual boilers. A semi-quantitative investigation into some geometrical modifications on the rectangular jets was also conducted at velocity ratios of λ=0, 0.55 and 1.4. The rectangular nozzles were fitted with base plates orientated at 90 degrees and 60 degrees to the direction of flow. The longitudinal flow visualisation study highlighted the effect of velocity ratio on the flow field of the primary and secondary jets. In particular it showed that the main structures of the primary and secondary jets are sensitive to the co-flowing conditions. The primary jet also experienced the formation of coherent structures close to the bluff body re-circulation region for λ>2.8. The quantitative transverse analysis of the rectangular jets showed that the primary jet and secondary jets close to the nozzle exit plane distorted with a change in co-flowing conditions. The primary jet experienced distortion for λ>1.4, and the secondary jets experienced distortion for λ <1.4. A plausible mechanism for this “distortion” can be explained by different co-flowing conditions altering the velocity gradients of the jet, thus changing the denomination of the counter rotating vortices present in the corners of rectangular jets, allowing them to alter jet shape. The transverse quantitative analysis of the rectangular jets allowed for graphical representation of the normalised concentration of the primary and secondary jets in the radial direction and the centreline mixture fraction decay. The analysis of the latter showed that the primary jet, under all co-flow conditions, reached self-similarity at approximately x/D =4, whereas the secondary jets did so at x/D =2. The primary jets observed greater rates of centreline dilution at high velocity ratios, whereas the secondary jets did so at λ=0.55. The quantification of the centreline concentration decay obeyed the inverse rate law for all co-flowing conditions. The first order decay constant K₁, was found to be heavily dependant on velocity ratio. The planar transverse quantitative data of the primary and secondary jets was used with the method of weighted squares to develop a regression model that would three-dimensionally reproduce the scalar mixing field as a function of velocity ratio. The regression model reproduces scalar quantities for λ=0 and λ=0.55 to 3.6 for the primary jet and λ=0.55 to 3.6 and ∞ for the secondary jet, and is capable of predicting primary and secondary bulk fluid concentrations within 30 to 40 % of the measured values. A sensitivity analysis on the regression model revealed that it is highly responsive to the momentum-controlling region between the jets with a change in velocity ratio. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1297627 / Thesis (Ph.D.) -- University of Adelaide, School of Chemical Engineering, 2007

Large scale power generation commences with the combustion of coal or other fuel, which in turn converts high pressure water into steam which then drives a turbine thus generating electricity. Burning high moisture coal, such as lignite, for power generation implies that a significant amount of energy is wasted in vaporising the moisture, which could otherwise be used in the steam raising process. This implies that more moist coal would be required to drive the same process than if the coal was drier, thus increasing the amount of combustion products such as greenhouses gases. Introducing a dried coal in an existing boiler will significantly change the heat flux profiles, which could result in boiler damage or excessive fouling. Flame temperature is influenced by the supply of reactants; in most cases the limiting reactant will be oxygen. The supply of oxygen (through air) to a pneumatically transported coal stream and subsequent reaction is controlled by the localised fluid mechanics or ‘mixing’. This research aims to provide an understanding of the mixing process between the pneumatically transported coal and air in brown coal fired boilers by modelling the individual jets. The effects of the change in velocity ratio for the air (secondary) jets and fuel (primary) jets of rectangular burners typical of those found in brown coal fired boilers has been studied experimentally and is reported in this thesis. In particular, scientific analysis was used to investigate the physical mechanisms which control fuel-air mixing, and to quantify the concentration of primary and secondary fluid. The concentration data was used in a regression model in conjunction with a reactive combustion model, developed from a 1:30 scale cold model of the Yallourn W’ stage 2 boiler, in order that overall boiler performance can be assessed. This overall study is fundamental as a result of the questions raised concerning the future of brown coal in modern society. A qualitative flow visualisation study of the unconfined 1:30 scaled primary, and two adjacent rectangular jets, was conducted using single colour planar laser induced fluorescence. The characteristics of the jet flow were examined by imaging individually seeded primary and secondary jets and were visualised through four different planes longitudinally, on the axes of each jet. In addition, a transverse qualitative and quantitative study on the rectangular jets was also conducted for the individually seeded jets, and was visualised through planes of flow perpendicular to the direction flow, specifically at axial stations of x/D =0.1, 0.2, 0.5, 1, 2, 4, 6 and 8. The flow characteristics were also examined under different co-flow conditions, particularly secondary to primary jet velocity ratios (λ) of 0, 0.55, 1.4, 2.8, 3.6 and ∞. This quantitative data yields the basis for a 3D regression model to predict fuel-air mixing in actual boilers. A semi-quantitative investigation into some geometrical modifications on the rectangular jets was also conducted at velocity ratios of λ=0, 0.55 and 1.4. The rectangular nozzles were fitted with base plates orientated at 90 degrees and 60 degrees to the direction of flow. The longitudinal flow visualisation study highlighted the effect of velocity ratio on the flow field of the primary and secondary jets. In particular it showed that the main structures of the primary and secondary jets are sensitive to the co-flowing conditions. The primary jet also experienced the formation of coherent structures close to the bluff body re-circulation region for λ>2.8. The quantitative transverse analysis of the rectangular jets showed that the primary jet and secondary jets close to the nozzle exit plane distorted with a change in co-flowing conditions. The primary jet experienced distortion for λ>1.4, and the secondary jets experienced distortion for λ <1.4. A plausible mechanism for this “distortion” can be explained by different co-flowing conditions altering the velocity gradients of the jet, thus changing the denomination of the counter rotating vortices present in the corners of rectangular jets, allowing them to alter jet shape. The transverse quantitative analysis of the rectangular jets allowed for graphical representation of the normalised concentration of the primary and secondary jets in the radial direction and the centreline mixture fraction decay. The analysis of the latter showed that the primary jet, under all co-flow conditions, reached self-similarity at approximately x/D =4, whereas the secondary jets did so at x/D =2. The primary jets observed greater rates of centreline dilution at high velocity ratios, whereas the secondary jets did so at λ=0.55. The quantification of the centreline concentration decay obeyed the inverse rate law for all co-flowing conditions. The first order decay constant K₁, was found to be heavily dependant on velocity ratio. The planar transverse quantitative data of the primary and secondary jets was used with the method of weighted squares to develop a regression model that would three-dimensionally reproduce the scalar mixing field as a function of velocity ratio. The regression model reproduces scalar quantities for λ=0 and λ=0.55 to 3.6 for the primary jet and λ=0.55 to 3.6 and ∞ for the secondary jet, and is capable of predicting primary and secondary bulk fluid concentrations within 30 to 40 % of the measured values. A sensitivity analysis on the regression model revealed that it is highly responsive to the momentum-controlling region between the jets with a change in velocity ratio. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1297627 / Thesis (Ph.D.) -- University of Adelaide, School of Chemical Engineering, 2007