Liebhart, Ewlad
(2000)
*Particle deposition and cake formation in filters with shearing flows.*
(PhD thesis), Kingston University,
.

## Abstract

Existing theories for the mechanical response of particle fluid mixtures have been reviewed and extended. They are made appropriate to geometries in which the dominant loading is a shearing one. The theories are then applied to the description of filtration experiments (these were performed by researchers dn a parallel research programme at Loughborough University). Two limits are distinguished: one in which particles experience a strong double layer interaction and one in which the particles are neutral and the fluid is the only significant force-mediating medium. The existing theories that have been reviewed and used are the quasi-static two-phase continuum mechanics framework (including seepage effects), the granular temperature theory for neutral particles and the common consolidation theory for strongly interacting particles. To extend these general theories - and especially to enter reliable constitutive relations - a micromechanical analysis is carried out and methods are developed to arrive at expressions for bulk properties. An analysis is performed of the response to a small localised fluctuation in either stress or solidosity of a particle-fluid mixture under arbitrary mean loading conditions. This analysis leads to a condition for stability of a mixture in terms of the solidosity sensitivity of the particle pressure and the solidosity sensitivity of the viscous constitutive parameters of the mixture given a mean loading regime. In the analysis it is recognised that a slurry in motion (especially shear) will always experience fluctuations. Two applications of the stability analysis are then presented. First it is recognised that homogenisation is impossible when the system is unstable. Second the border between a stable (packed) region and a free flowing region is defined by the edge of the stability condition, as made appropriate to the prevailing loading conditions. This piece of fundamental analysis is then used to describe filtration experiments, notably ones in which shear plays a distinctive role - these are torsion shear filtration and crossflow filtration. In order to analyse torsion shear filtration a calculation is carried out of a Newtonian fluid in a cylindrical vessel, loaded at the top by a rotating piston. A range of result is obtained: flow in an infinite cylinder, flow in a cylinder of finite length and flow in a finite cylinder with two immiscible fluids, occupying various sections of the cylindrical domain. The latter problem is particularly relevant to the torsion shear filtration problem as it shows that no significant shearing stress reaches, the cake until the fluid region near the piston is of the order of magnitude of the particle size of the mixture. Once shear can penetrate the cake the effects of it are noticed in that in a stable heterogeneous medium structures formation takes place in the direction of the major principal stress, implying that the greater the shear that is applied the greater the angle at which structures form. Then a calculation is presented to demonstrate the reduction in uniaxial stiffnes due to structures formation and the experimental result is recovered that for neutral particles cakes becoine denser when the shear is increased. This result is qualitative, though quantitative formulas are presented. The latter require parameters such as an estimate of the magnitude of the stiffness fluctuations that are hard to determine from current experiments. For double layer interacting particles the effects of shear are noticed at an earlier stage in the filtration process as particle interactions transmit the forces exerted externally on the mixture. The overall stiffness due to shearing is then estimated (stability is here required) and it is shown that the normal stress on the medium is reduced due to the fluctuations induced by the shearing. A lattice-Boltzmann, simulation of the same configuration confirms this interesting result. A crossflow setup has been analysed. A somewhat simplified one dimensional investigation is presented. The key point is that the edge of the cake near the septum is defined by the edge of stability analysis and this piece of information enables' a full survey of experimental results with a wide range of process paraméters (feed solidosity, crossflow velocity, crossflow pressure, particle type, pH). Two key experimental parameters are predicted: the end of filtration filtrate flow and time constant with which this end stage is reached. Double layer interacting particles and neutral particles have been explored. Some key findings pertaining especially to cases of thisn cakes are as follows. Double layer interacting particles: the end of filtration filtrate flux depends on the ratio of the crossflow velocity and feed solidosity only. The time constant depends Existing theories for the mechanical response of particle fluid mixtures have been reviewed and extended. They are made appropriate to geometries in which the dominant loading is a shearing one. The theories are then applied to the description of filtration experiments (these were performed by researchers dn a parallel research programme at Loughborough University). Two limits are distinguished: one in which particles experience a strong double layer interaction and one in which the particles are neutral and the fluid is the only significant force-mediating medium. The existing theories that have been reviewed and used are the quasi-static two-phase continuum mechanics framework (including seepage effects), the granular temperature theory for neutral particles and the common consolidation theory for strongly interacting particles. To extend these general theories - and especially to enter reliable constitutive relations - a micromechanical analysis is carried out and methods are developed to arrive at expressions for bulk properties. An analysis is performed of the response to a small localised fluctuation in either stress or solidosity of a particle-fluid mixture under arbitrary mean loading conditions. This analysis leads to a condition for stability of a mixture in terms of the solidosity sensitivity of the particle pressure and the solidosity sensitivity of the viscous constitutive parameters of the mixture given a mean loading regime. In the analysis it is recognised that a slurry in motion (especially shear) will always experience fluctuations. Two applications of the stability analysis are then presented. First it is recognised that homogenisation is impossible when the system is unstable. Second the border between a stable (packed) region and a free flowing region is defined by the edge of the stability condition, as made appropriate to the prevailing loading conditions. This piece of fundamental analysis is then used to describe filtration experiments, notably ones in which shear plays a distinctive role - these are torsion shear filtration and crossflow filtration. In order to analyse torsion shear filtration a calculation is carried out of a Newtonian fluid in a cylindrical vessel, loaded at the top by a rotating piston. A range of result is obtained: flow in an infinite cylinder, flow in a cylinder of finite length and flow in a finite cylinder with two immiscible fluids, occupying various sections of the cylindrical domain. The latter problem is particularly relevant to the torsion shear filtration problem as it shows that no significant shearing stress reaches, the cake until the fluid region near the piston is of the order of magnitude of the particle size of the mixture. Once shear can penetrate the cake the effects of it are noticed in that in a stable heterogeneous medium structures formation takes place in the direction of the major principal stress, implying that the greater the shear that is applied the greater the angle at which structures form. Then a calculation is presented to demonstrate the reduction in uniaxial stiffnes due to structures formation and the experimental result is recovered that for neutral particles cakes becoine denser when the shear is increased. This result is qualitative, though quantitative formulas are presented. The latter require parameters such as an estimate of the magnitude of the stiffness fluctuations that are hard to determine from current experiments. For double layer interacting particles the effects of shear are noticed at an earlier stage in the filtration process as particle interactions transmit the forces exerted externally on the mixture. The overall stiffness due to shearing is then estimated (stability is here required) and it is shown that the normal stress on the medium is reduced due to the fluctuations induced by the shearing. A lattice-Boltzmann , simulation of the same configuration confirms this interesting result. A crossflow setup has been analysed. A somewhat simplified one dimensional investigation is presented. The key point is that the edge of the cake near the septum is defined by the edge of stability analysis and this piece of information enables' a full survey of experimental results with a wide range of process paraméters (feed solidosity, crossflow velocity, crossflow pressure, particle type, pH). Two key experimental parameters are predicted: the end of filtration filtrate flow and time constant with which this end stage is reached. Double layer interacting particles and neutral particles have been explored. Some key findings pertaining especially to cases of thisn cakes are as follows. Double layer interacting particles: the end of filtration filtrate flux depends on the ratio of the crossflow velocity and feed solidosity only. The time constant depends quadratically on the cake thickness and is inversely proportional to the crossflow velocity. Experimental results for the end of filtration filtrate flux show very good agreement with the theoretical predictions; for the time constant a more qualitative agreement can be confirmed as it is difficult to ascertain what the cake thickness is in each of the experiments. Neutral particles: granular temperature theory has been deployed with great success. An interesting effect - one that is also demonstrated by experiments - is that the crossflow velocity dependence of the end of filtration filtrate flux is a function of the particle size (in relation to the conduit size). For large particles the mean free path is such that Knudsen conditions prevail. In this case the collisions with the walls of the conduit dominate over the interparticle interactions and continuity of particle pressure together with the edge of stability criterion yield the prediction that an increased value for the crossflow velocity leads to a decreased value for the end of filtration filtrate flux. For small particles (where interparticle interactions dominate) the opposite is found. The time constant of the process is also predicted for thin cakes and is shown to decrease for increasing crossflow velocities. This result is again borne out by the relevant experiments.

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