(266b) Particle-Pressure Induced Transient Pore Pressure Changes in Oscillated Particle-Fluid Systems

Experiments on oscillated particle-fluid systems in a dead-end filtration configuration reveal a rich variety in behaviour, Gundogdu et al (2003^a). In these experiments the septum at the bottom of the apparatus is vibrated normally, while a constant fluid pressure is maintained across a densely packed particle bed (permeation experiment). The filtration rate is recorded as a function of the oscillation acceleration amplitude, see Figure 1. At small amplitudes the filtration rate drops ? indicating either densification of the packed bed or pressure drop reduction. At a distinct amplitude the particulate material fluidizes and a step-change in the filtration rate is observed, as the septum is unclogged. This critical point is obviously of great interest to design engineers and research has been directed predominantly at understanding both its occurrence (Gundogdu et al (2003^a)) and the behaviour of the fluidized slurry at vibration amplitudes that are greater than the critical value (Gundogdu et al (2003^b)). There is also much to be learnt about particle fluid systems from the behaviour below the critical point, when the particulate medium remains densely packed and this régime is the topic of this paper.

For densely packed particulate media a constitutive law needs to be put forward to describe the behaviour of stress and strain under cyclic loading. Clearly, a stress-path dependence is evident. Much work has been reported in the civil engineering literature on deviatorically cycled specimens of granular material, for example Bouckovalas et al (1984). For the problem to hand a compressive/expansive path needs to be considered. The convex nature of the static stress-strain law (i.e. the material becomes stiffer when it is more compressed), coupled to the requirement that unloading must be stiffer than loading to satisfy positive damping requirements, has certain implications. These are demonstrated graphically in Figure 2. A strain-driven system is considered. Ignoring any residual stress evolution (which will vanish after many cycles) the stress-strain path is demonstrated. The unloading curve is initially stiffer than the loading, which gives the unloading curve its shape. The stress path associated with the cycle is an oscillatory component with an amplitude, as shown. At the same time there must be a mean stress to accommodate the convex character of the curve. The difference between the mean fluctuating stress and the average stress is the particle pressure for a packed bed. It is an expansive stress. Thus, bed that is cyclically loaded with a compressive/expansive strain cycle develops an expansive particle pressure as well as a cyclic stress cycle. The same does not happen in a strain-softening material, such as a granular aggregate that is loaded deviatorically. A particle pressure does develop for a fluidized slurry that is vibrated, see Gundogdu et al (2003^b). For the test discussed above the strain cycle is compressive/expansive.

Figure 1: Filtrate flow rate as a function of the vibration acceleration amplitude in dead-end vibrated permeation experiments. From Gundogdu et al (2003^a)

Figure 2: Sketch of the stress-strain curves relevant to a strain-controlled cyclic loading experiment.

The implications of the secular particle pressure in a fluid environment are interesting and further explored. Two analyses are presented that correspond to concurrent phenomena. First, the strain amplitude propagates through the medium; this phenomenon, which shows that the oscillatory behaviour is localised near the septum, is described by solving the appropriate wave equation approximated for small amplitude. The analysis is similar to the one presented by Gundogdu et al (2003^a), though a refinement for compressible fluids is introduced. Second, the secular particle pressure, associated with the oscillatory motion, has an impact on the distribution of fluid and solid stresses in the medium. Broadly speaking the effect works as follows: the total stress remains the same; the particle pressure, therefore, needs to be compensated by the fluid pressure. Therefore, as the particle particle pressure near the septum goes up due to the oscillation, so the fluid pressure in that location comes down. The latter effect reduces the flow through the filter to the extent found in the experiments. The change in the flow needs time to consolidate and as a result the change in the pressure field has a transient component. The latter is modelled and calculated and presented in terms of the system variables: particle size, porosity and externally applied hydraulic gradient.

References

R. Bouckovalas, R.V. Whitman and W.A. Marr (1984) Permanent displacement of sand with cyclic loading. J. Geotechnical Eng. 110 (11), 1606-1623.

O. Gundogdu, M.A. Koenders, R.J. Wakeman, P. Wu. (2003^a) Permeation Through a Bed on a Vibrating Medium:Theory and Experimental Results. Chem Eng Science 58 (9), 1703-1713.

O. Gundogdu, M.A. Koenders, R.J. Wakeman, P. Wu. (2003^b) Vibration-assisted dead-end filtration: experiments and theoretical concepts. Chemical Engineering Research and Design: 81 (8) 916-923.