William B. Zimmerman

Nonlinear viscous fingering in miscible displacement with anisotropic dispersion

The effect of anisotropic dispersion on nonlinear viscous fingering in miscible displacements is examined. The formulation admits dispersion coefficient‐velocity field couplings (i.e., mechanical dispersivities) appropriate to both porous media and Hele–Shaw cells. A Hartley transform‐based scheme is used to numerically simulate unstable miscible displacement. Several nonlinear finger interactions were observed. Shielding, spreading, tip splitting, and pairing of viscous fingers were observed here, as well as in isotropic simulations. Multiple coalescence and fading were observed in simulations with weak lateral dispersion, but not for isotropic dispersion. Transversely and longitudinally averaged one‐dimensional concentration histories demonstrate the rate at which the mixing zone broadens and the increase in lateral scale as the fingers evolve when no tip splitting occurs. These properties are insensitive to both the dispersion anisotropy and the Peclet number at high Peclet number and long times. This suggests the dominance of finger interaction mechanisms that are essentially independent of details of the concentration fields and governed fundamentally by pressure fields.

Viscous fingering in miscible displacements: Unification of effects of viscosity contrast, anisotropic dispersion, and velocity dependence of dispersion on nonlinear finger propagation

The two‐dimensional (2‐D) isotropic simulations of Tan and Homsy [Phys. Fluids 31, 1330 (1988)] are extended to much broader and longer domains, and the 2‐D anisotropic simulations of Zimmerman and Homsy are extended to include a general velocity dependence. The mechanisms of nonlinear interaction of viscous fingers found for the first time in the anisotropic simulations recur in isotropic simulations, but at weaker levels of dispersion. An appropriate scaling to unify the average long time growth of the instability with both anisotropy in geometry and dispersion is provided. The long time growth of the instability from simulations agrees with acoustic measurements in 3‐D porous media, Bacri et al. [Phys. Rev. Lett. 67, 2005 (1991)], elucidating the effects of viscosity contrast, anisotropy, and velocity dependence of longitudinal dispersion. The combination of sufficiently high viscosity contrast, weak transverse dispersion, and strong dependence of longitudinal dispersion on velocity results in an augme...

Direct flow-pattern identification using electrical capacitance tomography

Abstract Non-invasive techniques such as process tomography are beginning to make promising contributions to control systems and are well fitted for flow pattern identification in opaque pipes or conduits. Even though process tomography and electrical capacitance tomography are usually associated with imaging, the image in itself can very seldom be directly used for control purpose. The path to be followed from data collection to flow-pattern identification often involves an image reconstruction phase, followed by some form of image processing and analysis. When online control is concerned, this approach may be too time consuming. This paper proposes a direct approach that discriminates between an annular and stratified flow pattern without the need for imaging. The model relies on capacitance data collected from an experimental rig, and extracts the information content from these raw-data measurements. Flow-pattern identifiers are kept simple and are designed to be rapidly evaluated for online observation purpose. Additional information and monitoring, such as the phase distribution or the detection of a slug building process, can also be obtained from the data by using specific electrode pair combinations.

Three-dimensional viscous fingering : a numerical study

Multidimensional viscous fingering is accurately simulated by an extension of the Hartley spectral methods to three dimensions. Two types of initial conditions were used in the rectilinear displacements−white noise and polygonal cells of the wave number of the mode of maximum instability identified by the linear theory. White noise initial conditions demonstrated that the mechanisms of nonlinear interaction of viscous fingers found in two‐dimensional (2‐D) simulations persist to three dimensions. Further, the long‐time rate of advance of viscous fingers remains unchanged from two dimensions, suggesting that 2‐D simulations are sufficient to capture essential features of nonlinear viscous fingering. Simulations with polygonal cellular symmetries, specifically square cells and hexagonal cells, illustrated that weak nonlinear theory cannot predict the shape selection of viscous fingers, as the mechanisms of finger interactions that dominate shape selection lack transverse symmetry and symmetry with respect t...

Kinematic model of bubble motion in a flotation froth

Abstract A kinematic model for bubble motion in a flotation froth is proposed. The model describes bubble trajectories in a two-dimensional vertical section of the froth phase of a flotation cell. If the viscosity and permeability of the froth is constant over the field, then the motion of bubbles is irrotational and Laplaces equation can be used to describe the setamline of the bubble motion in terms of the air flux across the boundaries of the rectangular region. The justification for this kinematic solution is based on the computation of isobars in the froth using Darcys law. This procedure offers the possibility of future extension to the more general case where the viscosity and permeability of the froth varies with position. Some examples of the dependence of the streamline pattern on the boundary conditions, which in turn can represent non-uniform distributions of air entering and leaving the region, are given. In particular, the permeability of the upper boundary, which corresponds to the top surface of the froth, is used to model the bursting rate of the bubbles. The unburst bubbles are assumed to cross the overflow weir in the side wall as part of the concentrate flow. The kinematic bubble trajectories will be used in a computer simulation of the dynamic behaviour of an experimental flotation test. It is intended that validation of the model will be used on comparison of visual output of the simulation with video records of experimental tests.

Fluidic oscillator-mediated microbubble generation to provide cost effective mass transfer and mixing efficiency to the wastewater treatment plants.

Aeration is one of the most energy intensive processes in the waste water treatment plants and any improvement in it is likely to enhance the overall efficiency of the overall process. In the current study, a fluidic oscillator has been used to produce microbubbles in the order of 100 μm in diameter by oscillating the inlet gas stream to a pair of membrane diffusers. Volumetric mass transfer coefficient was measured for steady state flow and oscillatory flow in the range of 40-100l/min. The highest improvement of 55% was observed at the flow rates of 60, 90 and 100l/min respectively. Standard oxygen transfer rate and efficiency were also calculated. Both standard oxygen transfer rate and efficiency were found to be considerably higher under oscillatory air flow conditions compared to steady state airflow. The bubble size distributions and bubble densities were measured using an acoustic bubble spectrometer and confirmed production of monodisperse bubbles with approximately 100 μm diameters with fluidic oscillation. The higher number density of microbubbles under oscillatory flow indicated the effect of the fluidic oscillation in microbubble production. Visual observations and dissolved oxygen measurements suggested that the bubble cloud generated by the fluidic oscillator was sufficient enough to provide good mixing and to maintain uniform aerobic conditions. Overall, improved mass transfer coefficients, mixing efficiency and energy efficiency of the novel microbubble generation method could offer significant savings to the water treatment plants as well as reduction in the carbon footprint.

Effects of CO 2 and pH on Growth of the Microalga Dunaliella salina

A potentially cost-effective and scalable method to stabilize pH in microalgal batch cultures is proposed in this study. The cultures were supplied with different concentrations of CO2 enriched gas and controlled amounts of bicarbonate were added. An empirical model correlating the equilibrium pH to bicarbonate and CO2 stream concentrations was established experimentally. Finally, the isolated impact of either pH or CO2 concentration on Dunaliella salina growth was studied.

The diffusion coefficient of a swollen microgel particle

The drag on a permeable particle traversing through a Newtonian liquid is calculated. This is in terms of a single dimensionless group, Da, the Darcy number, which relates the particle permeability to the radius. For small values of the Darcy number the solution reverts to the well-known Stokes drag for smooth hard particles. For larger values of the Darcy number the drag is reduced. This drag allows calculation of the diffusion coefficient of such particles, again as a function of the Darcy number. The results are discussed in relation to microgel particles, which display swelling under certain conditions. The size of such particles is typically measured with dynamic light scattering, which measures the diffusion coefficient of particles and as such the analysis presented here shows the conditions under which dynamic light scattering is accurate. The Darcy number for microgel particles is estimated to be on the order of 10(-7).

On slip velocity boundary conditions for electroosmotic flow near sharp corners

The Helmholtz–Smoluchowski (HS) slip velocity boundary condition is often used in computational models of microchannel flows because it allows the motion of the electric double layer (EDL) to be approximated without resolving the charge density profiles close to the walls while dramatically reducing the computational effort required to solve the flow model. The approximation works well for straight channel flows but breaks down in areas of high wall curvature such as sharp corners, where large nonphysical velocities are generated. Many microfluidic applications such as the on-chip focusing and separation of biomolecules rely on the interaction of electroosmosis and electrophoresis in complex channel geometries. In order for these effects to be properly treated using the slip velocity boundary condition, the errors introduced into the solution at corners must be understood. In this article, a complete model for the ion concentrations, electric field, and fluid flow in complex microchannel geometries is pre...

Fast binary reactions in a heterogeneous catalytic batch reactor

When heterogeneous chemical reaction is su8ciently fast, transport of reactants becomes limiting. In a well stirred, batch reactor, macroscopic concentration gradients can be eliminated as a factor limiting the rate of reaction, leaving only the mesoscopic mass transfer of reactants to the surface of the catalyst as limiting, if the reaction does not occur inside a porous support. Here, a transformation of the governing equations for the time-dependence of bulk and surface concentrations results in second order ODE in time and a single nonlinear constraint with boundary values at the initial and in=nite times for two auxiliary variables termed modi=ed Thiele moduli. This system of two equations—one diAerential, one algebraic—and two unknowns is an exact consequence of the governing equations (three ODEs and three algebraic constraints). The power of this formulation is demonstrated by analytic solutions for irreversible and nearly irreversible theories. These solutions are corroborated by full nonlinear numerical computations of the boundary value problem, for the case when asymmetric mass transfer coe8cients admit the possibility that the mode of operation switches from relative surface depletion of one reactant to depletion of the other in a binary reaction. The modi=ed Thiele modulus formulation reveals the time scale for the switch over, as well as giving a reliable prediction for the time scale for 99% conversion based on the switch time identi=ed from the irreversible theory. ? 2003 Elsevier Science Ltd. All rights reserved.