In order to develop a better understanding of the role of the froth phase in coal flotation, froth stability measurements have been conducted using a continuous flotation cell. The results indicate that the mass flow rate of coal or mineral matter reporting to the product is linearly related to the flow rate of water reporting to the product. This relationship has been used to distinguish the relative contributions of hydrophobic attachment or hydraulic entrainment to the total product recovery. Simple mathematical expressions have also been developed to characterize the cleanability of various coals. A mathematical model based on first principles has been developed to provide additional insight into the complex relationship between the various operating parameters of a flotation cell and the froth behavior. The predicted results compare favorably to actual flotation test data for most conditions. In addition, a simple method was developed to study the fundamental interactions of various frothing agents with coal particles. The results of these studies indicate that the adsorption behavior of frothers in the presence of coal depends on both the physical properties of the frothing agent and the coal. / Master of Science

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Thesis. 1976. M.S.--Massachusetts Institute of Technology. Dept. of Mechanical Engineering. / Microfiche copy available in Archives and Engineering. / Bibliography: p.163-164. / M.S.

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The nonlinear dynamic characteristics and stability of floating vehicles are investigated theoretically and experimentally. Mathematical models of such floating bodies are used to investigate their complicated motions in regular waves. In particular, we address the phenomenon of indirectly exciting the roll motion of a vessel due to nonlinear couplings of the heave, pitch, and roll modes. In the analytical approach the method of multiple scales is used to determine first-order approximations to the solutions, yielding a system of nonlinear first-order equations governing the modulation of the amplitudes and phases of the system. The fixed-point solutions of these equations are determined and their bifurcations are investigated. Hopf bifurcations are found. Numerical simulations are used to investigate the bifurcations of the ensuing limit cycles and how they produce chaos. Experiments are conducted with tanker and destroyer models. They demonstrate some of the nonlinear effects, such as the jump phenomenon, the subcritical instability, and the coexistence of multiple solutions. The experimental results are qualitatively in good agreement with the results predicted by the theory. Coupling of the pitch and roll motions of a vessel when their frequencies are in the ratio of two-to-one is modeled by a two-degree-of-freedom system. The damping in the pitch mode is modeled by a linear viscous damping, whereas that of the roll mode is modeled by the sum of a linear viscous part and a quadratic viscous part. The effect of the quadratic damping is investigated when either mode is externally excited through a primary resonance. Force-response and frequency-response curves are generated. Coexistence of multiple solutions is found. The jump phenomenon continues to exist, whereas the saturation phenomenon ceases in the presence of quadratic damping. Hopf bifurcations are found. They indicate conditions for the nonexistence of steady-state periodic responses. Instead, the response is an amplitude- and phase-modulated motion consisting of both modes. Floquet theory is used to determine the stability of limit-cycle solutions. They undergo a pitchfork bifurcation followed by a cascade of period-doubling bifurcations, leading to chaos and hence chaotically modulated motions. When the roll mode is excited, the quadratic damping causes the region between the two Hopf bifurcation frequencies to shrink. However, the quadratic damping which may be introduced by attaching antirolling devices does not eliminate complicated motions completely in this region. The dynamic stability and excessive motion of the roll mode of a vessel in following or head regular waves is investigated theoretically and experimentally. The motion is modeled by a three-degree-of-freedom system with quadratic and cubic nonlinearities. The heave and pitch modes are linearized and their harmonic solutions are coupled into the roll mode. The resulting nonlinear ordinary-differential equation with time-varying coefficients is used to determine the stability of the roll mode for the case of principal parametric resonance. Experiments with a tanker model were conducted to validate the theory. They demonstrate the jump phenomenon and subcritical instability. They also reveal that the large-amplitude roll response depends not only on the encounter frequency but also on the position of the model relative to the waves. / Ph. D.

This thesis investigates wave loading on bodies in the free surface using smoothed particle hydrodynamics (SPH). This includes wave loading on fixed bodies, waves generated by heaving bodies in still water and the heave response of a body in waves, representing a wave energy device. SPH is a flexible Lagrangian technique for CFD simulations, which in principle applies to steep and breaking waves without special treatment allowing us to simulate highly nonlinear and potentially violent flows encountered in a real sea. However few detailed tests have been undertaken even with small amplitude waves.This research uses the open-source SPH code SPHysics. First two forms of SPH formulation, standard SPH with artificial viscosity and SPH-Arbitrary Lagrange Euler (ALE) with a Riemann solver, are used to simulate progressive waves in a 2-D tank. The SPH-ALE formulation with a symplectic time integration scheme and cubic spline kernel is found to model progressive waves with negligible dissipation whereas with the standard SPH formulation waves decay markedly along the tank. We then consider two well-defined test cases in two dimensions: progressive waves interacting with a fixed cylinder and waves generated by a heaving semi-immersed cylinder. To reduce computer time in a simple manner a variable particle mass distribution is tested with fine resolution near the body and coarse resolution further away, while maintaining a uniform kernel size. A mass ratio of 1:4 proved effective but increasing to 1:16 caused particle clumping and instability. For wave loading on a half-submerged cylinder the agreement with the experimental data of Dixon et al. (1979) for the root mean square force is within 2%. For more submerged cases, the results show some discrepancy, but this was also found with other modelling approaches. For the heaving cylinder, SPH results for the far field wave amplitude and vertical force on the cylinder show good agreement with the data of Yu and Ursell (1961). The variable mass distribution leads to a computer run time speedup of nearly 200% in these cases on a single CPU. The results of the vertical force and wave amplitude are shown to be quite sensitive to the value of the slope limiter in the Riemann solver for the 2-D heaving cylinder problem. A heaving 2-D wedge or 3-D cone whose oscillatory vertical motion is prescribed as the elevation of a focused wave group is a precise test case for numerical free-surface schemes. We consider two forms of repulsive boundary condition (Monaghan & Kos, 1999, and Rogers et al., 2008) and particle boundary force (Kajtar and Monaghan, 2009) for the 2-D wedge case, comparing the result with the experimental data of Drake et al. (2009). The repulsive boundary condition was more effective than the particle boundary force method. Variable particle mass with different kernel sizes was then tested for 2-D problems for mass ratios of 1:4, 1:16 and 1:4:16 with satisfactory results without particle clumping and instability. For the 3-D cone case, SPH reproduces the experimental results very closely for the lower frequency tested where there is no separation from the bottom surface of the body but for the higher frequencies the magnitudes of force minima were underestimated. The mass ratios of 1:8 and 1:8:27 in two and three nested regions are tested for the 3-D cone problem where a computer run time speedup of nearly 500% is achieved on 16 processors for the mass ratio of 1:8.Finally, the floating body of a heaving wave energy device known as the Manchester Bobber is modelled in extreme waves without power take-off. The results for a single float are in approximate agreement with the experiment.