Mathematical Modelling of the Dynamics of Granular Materials in a Rotating Cylinder

The current study is aimed at the development of a theoretical simulation tool based on Discrete Element Method (DEM) to 'interpret granular dynamics of solid bed in the cross section of the horizontal rotating cylinder at the microscopic level and subsequently apply this model to establish the transition behaviour, mixing and segregation.The simulation of the granular motion developed in this work is based on solving Newton's equation of motion for each particle in the granular bed subjected to the collisional forces, external forces and boundary forces. At every instant of time, the forces are tracked and the positions velocities and accelarations of each partcle is The software code for this simulation is written in VISUAL FORTRAN 90 After checking the validity of the code with special tests, it is used to investigate the transition behaviour of granular solids motion in the cross section of a rotating cylinder for various rotational speeds and fill fraction.This work is hence directed towards a theoretical investigation based on Discrete Element Method (DEM) of the motion of granular solids in the radial direction of the horizontal cylinder to elucidate the relationship between the operating parameters of the rotating cylinder geometry and physical properties ofthe granular solid.The operating parameters of the rotating cylinder include the various rotational velocities of the cylinder and volumetric fill. The physical properties of the granular solids include particle sizes, densities, stiffness coefficients, and coefficient of friction Further the work highlights the fundamental basis for the important phenomena of the system namely; (i) the different modes of solids motion observed in a transverse crosssection of the rotating cylinder for various rotational speeds, (ii) the radial mixing of the granular solid in terms of active layer depth (iii) rate coefficient of mixing as well as the transition behaviour in terms of the bed turnover time and rotational speed and (iv) the segregation mechanisms resulting from differences in the size and density of particles.The transition behaviour involving its six different modes of motion of the granular solid bed is quantified in terms of Froude number and the results obtained are validated with experimental and theoretical results reported in the literature The transition from slumping to rolling mode is quantified using the bed turnover time and a linear relationship is established between the bed turn over time and the inverse of the rotational speed of the cylinder as predicted by Davidson et al. [2000]. The effect of the rotational speed, fill fraction and coefficient of friction on the dynamic angle of repose are presented and discussed. The variation of active layer depth with respect to fill fraction and rotational speed have been investigated. The results obtained through simulation are compared with the experimental results reported by Van Puyvelde et. at. [2000] and Ding et at. [2002].The theoretical model has been further extended, to study the rmxmg and segregation in the transverse direction for different particle sizes and their size ratios. The effect of fill fraction and rotational speed on the transverse mixing behaviour is presented in the form of a mixing index and mixing kinetics curve. The segregation pattern obtained by the simulation of the granular solid bed with respect to the rotational speed of the cylinder is presented both in graphical and numerical forms. The segregation behaviour of the granular solid bed with respect to particle size, density and volume fraction of particle size has been investigated. Several important macro parameters characterising segregation such as mixing index, percolation index and segregation index have been derived from the simulation tool based on first principles developed in this work.

Influence of Angle of Air Injection and Particles in Bed Hydrodynamics of Swirling Fluidized Bed

The thesis presented here unveils an experimental study of the hydrodynamic characteristics of swirling fluidized bed viz. pressure drop across the distributor and the bed, minimum fluidizing velocity, bed behaviour and angle of air injection. In swirling fluidized bed the air is admitted to the bed at an angle 'Ѳ' to the horizontal. The vertical component of the velocity v sin Ѳ causes fluidization and the horizontal component v cos Ѳ contributes to swirl motion of the bed material.The study was conducted using spherical particles having sizes 3.2 mm, 5.5 mm & 7.4 mm as the bed materials. Each of these particles was made from high density polyethylene, nylon and acetal having relative densities of 0.93, 1.05 and 1.47 respectively.The experiments were conducted using conidour type distributors having four rows of slits. Altogether four distributors having angles of air injection (Φ)- 0°, 5°, 10° & 15° were designed and fabricated for the study. The total number of slits in each distributor was 144. The area of opening was 6220 mm2 making the percentage area of opening to 9.17. But the percentage useful area of opening of the distributor was 96.The experiments on the variation of distributor pressure drop with superficial velocity revealed that the distributor pressure drop decreases with angle of air injection. Investigations related to bed hydrodynamics were conducted using 2.5 kg of bed material. The bed pressure drop measurements were made along the radial direction of the distributor at distances of 60 mm, 90 mm, 120 mm & 150 mm from the centre of the distributor. It was noticed that after attaining minimum fluidizing velocity, the bed pressure drop increases along the radial direction of the distributor. But at a radial distance of 90 mm from the distributor centre, after attaining minimum fluidizing velocity the bed pressure drop remains almost constant. It was also observed that the bed pressure drop varies inversely with particle size as well as particle density.An attempt was made to determine the effect of various parameters on minimum fluidizing velocity. It was noticed that the minimum fluidizing velocity varies directly with angle of air injection (Φ), particle size and particle density.The study on the bed behaviour showed that the superficial velocity required for initiating various bed phenomena (such as swirl motion and separation of particles from the cone at the centre) increase with increase in particle size as well as particle density. It was also observed that the particle size and particle density directly influence the superficial velocity required for various regimes of bed behaviour such as linear variation of bed pressure drop, constant bed pressure drop and sudden increase or decrease in bed pressure drop.Experiments were also performed to study the effect of angle of air injection (Φ). It was noticed that the bed pressure drop decreases with angle of air injection. It was also noticed that the angle of air injection directly influence the superficial velocity required for initiating various bed phenomena as well as the various regimes of bed behaviour.

Acoustic radiation and scattering from cylindrical bodies and analysis of transducer arrays

New mathematical methods to analytically investigate linear acoustic radiation and scattering from cylindrical bodies and transducer arrays are presented. Three problems of interest involving cylinders in an infinite fluid are studied. In all the three problems, the Helmholtz equation is used to model propagation through the fluid and the beam patterns of arrays of transducers are studied. In the first problem, a method is presented to determine the omni-directional and directional far-field pressures radiated by a cylindrical transducer array in an infinite rigid cylindrical baffle. The solution to the Helmholtz equation and the displacement continuity condition at the interface between the array and the surrounding water are used to determine the pressure. The displacement of the surface of each transducer is in the direction of the normal to the array and is assumed to be uniform. Expressions are derived for the pressure radiated by a sector of the array vibrating in-phase, the entire array vibrating in-phase, and a sector of the array phase-shaded to simulate radiation from a rectangular piston. It is shown that the uniform displacement required for generating a source level of 220 dB ref. μPa @ 1m that is omni directional in the azimuthal plane is in the order of 1 micron for typical arrays. Numerical results are presented to show that there is only a small difference between the on-axis pressures radiated by phased cylindrical arrays and planar arrays. The problem is of interest because cylindrical arrays of projectors are often used to search for underwater objects. In the second problem, the errors, when using data-independent, classical, energy and split beam correlation methods, in finding the direction of arrival (DOA) of a plane acoustic wave, caused by the presence of a solid circular elastic cylindrical stiffener near a linear array of hydrophones, are investigated. Scattering from the effectively infinite cylinder is modeled using the exact axisymmetric equations of motion and the total pressures at the hydrophone locations are computed. The effect of the radius of the cylinder, a, the distance between the cylinder and the array, b, the number of hydrophones in the array, 2H, and the angle of incidence of the wave, α, on the error in finding the DOA are illustrated using numerical results. For an array that is about 30 times the wavelength and for small angles of incidence (α<10), the error in finding the DOA using the energy method is less than that using the split beam correlation method with beam steered to α; and in some cases, the error increases when b increases; and the errors in finding the DOA using the energy method and the split beam correlation method with beam steered to α vary approximately as a7 / 4 . The problem is of interest because elastic stiffeners – in nearly acoustically transparent sonar domes that are used to protect arrays of transducers – scatter waves that are incident on it and cause an error in the estimated direction of arrival of the wave. In the third problem, a high-frequency ray-acoustics method is presented and used to determine the interior pressure field when a plane wave is normally incident on a fluid cylinder embedded in another infinite fluid. The pressure field is determined by using geometrical and physical acoustics. The interior pressure is expressed as the sum of the pressures due to all rays that pass through a point. Numerical results are presented for ka = 20 to 100 where k is the acoustic wavenumber of the exterior fluid and a is the radius of the cylinder. The results are in good agreement with those obtained using field theory. The directional responses, to the plane wave, of sectors of a circular array of uniformly distributed hydrophones in the embedded cylinder are then computed. The sectors are used to simulate linear arrays with uniformly distributed normals by using delays. The directional responses are compared with the output from an array in an infinite homogenous fluid. These outputs are of interest as they are used to determine the direction of arrival of the plane wave. Numerical results are presented for a circular array with 32 hydrophones and 12 hydrophones in each sector. The problem is of interest because arrays of hydrophones are housed inside sonar domes and acoustic plane waves from distant sources are scattered by the dome filled with fresh water and cause deterioration in the performance of the array.

Numerical Simulation of Two-Part Underwater Towing System

The motion instability is an important issue that occurs during the operation of towed underwater vehicles (TUV), which considerably affects the accuracy of high precision acoustic instrumentations housed inside the same. Out of the various parameters responsible for this, the disturbances from the tow-ship are the most significant one. The present study focus on the motion dynamics of an underwater towing system with ship induced disturbances as the input. The study focus on an innovative system called two-part towing. The methodology involves numerical modeling of the tow system, which consists of modeling of the tow-cables and vehicles formulation. Previous study in this direction used a segmental approach for the modeling of the cable. Even though, the model was successful in predicting the heave response of the tow-body, instabilities were observed in the numerical solution. The present study devises a simple approach called lumped mass spring model (LMSM) for the cable formulation. In this work, the traditional LMSM has been modified in two ways. First, by implementing advanced time integration procedures and secondly, use of a modified beam model which uses only translational degrees of freedoms for solving beam equation. A number of time integration procedures, such as Euler, Houbolt, Newmark and HHT-α were implemented in the traditional LMSM and the strength and weakness of each scheme were numerically estimated. In most of the previous studies, hydrodynamic forces acting on the tow-system such as drag and lift etc. are approximated as analytical expression of velocities. This approach restricts these models to use simple cylindrical shaped towed bodies and may not be applicable modern tow systems which are diversed in shape and complexity. Hence, this particular study, hydrodynamic parameters such as drag and lift of the tow-system are estimated using CFD techniques. To achieve this, a RANS based CFD code has been developed. Further, a new convection interpolation scheme for CFD simulation, called BNCUS, which is blend of cell based and node based formulation, was proposed in the study and numerically tested. To account for the fact that simulation takes considerable time in solving fluid dynamic equations, a dedicated parallel computing setup has been developed. Two types of computational parallelisms are explored in the current study, viz; the model for shared memory processors and distributed memory processors. In the present study, shared memory model was used for structural dynamic analysis of towing system, distributed memory one was devised in solving fluid dynamic equations.