scholarly journals Closed-form expression for the dynamic dispersion coefficient in Hagen-Poiseuille flow

2017 ◽  
Author(s):  
Lichun Wang ◽  
M. Bayani Cardenas
Keyword(s):  
Poiseuille Flow ◽  
Molecular Diffusion ◽  
Numerical Solutions ◽  
Dispersion Coefficient ◽  
Dispersion Model ◽  
Taylor Dispersion ◽  
Breakthrough Curves ◽  
Exact Expression ◽  
Closed Form Expression ◽  
Pore Network Model

We present an exact expression for the upscaled dynamic dispersion coefficient (D) for one-dimensional transport by Hagen-Poiseuille flow which is the basis for modeling transport in porous media idealized as capillary tubes. The theoretical model is validated by comparing the breakthrough curves (BTCs) from a 1D advection-dispersion model with dynamic D to that from direct numerical solutions utilizing a 2D advection-diffusion model. Both Taylor dispersion theory and our new theory are good predictors of D at lower Peclet Number (Pe) regime, but gradually fail to capture most parts of BTCs as Pe increases. However, our model generally predicts the mixing and spreading of solutes better than Taylor’s theory since it covers all transport regimes from molecular diffusion, through anomalous transport, and to Taylor dispersion. The model accurately predicts D based on the early part of BTCs even at relatively high Pe regime (~62) where the Taylor’s theory fails. Furthermore, the model allows for calculation of the time scale that separates Fickian from non-Fickian transport. Therefore, our model can readily be used to calculate dispersion through short tubes of arbitrary radii such as the pore throats in a pore network model.

Lagrangian approach to time-dependent laminar dispersion in rectangular conduits. Part 1. Two-dimensional flows

1988 ◽  
Vol 190 ◽  
pp. 201-215 ◽  
Author(s):  
Shimon Haber ◽  
Roberto Mauri
Keyword(s):  
Channel Flow ◽  
Poiseuille Flow ◽  
Open Channel ◽  
Molecular Diffusion ◽  
Dispersion Coefficient ◽  
Brownian Particle ◽  
Open Channel Flow ◽  
Taylor Dispersion ◽  
Time Dependent ◽  
Two Dimensional

Time-dependent mean velocities and dispersion coefficients are evaluated for a general two-dimensional laminar flow. A Lagrangian method is adopted by which a Brownian particle is traced in an artificially restructured velocity field. Asymptotic expressions for short, medium and long periods of time are obtained for Couette flow, plane Poiseuille flow and open-channel flow over an inclined flat surface. A new formula is suggested by which the Taylor dispersion coefficient can be evaluated from purely kinematical considerations. Within an error of less than one percent, over the entire time domain and for various flow fields, a very simple analytical expression is derived for the time-dependent dispersion coefficient \[ \tilde{D}(\tau) = D + D^T\left(1-\frac{1-{\rm e}^{-\alpha\tau}}{a\tau}\right), \] where D is the molecular diffusion coefficient, DT denotes the Taylor dispersion coefficient, τ stands for the non-dimensional time π2Dt/Y/, Y is the distance between walls and a = (N + 1)2 is an integer which is determined by the number of symmetry planes N that the flow field possesses. For Couette and open-channel flow there are no planes of symmetry and a = 1; for Poiseuille flow there is one plane of symmetry and a = 4.

Dispersion in steady and oscillatory flows through a tube with reversible and irreversible wall reactions

2005 ◽  
Vol 462 (2066) ◽  
pp. 481-515 ◽  
Author(s):  
Chiu-On Ng
Keyword(s):  
Dispersion Coefficient ◽  
Tube Wall ◽  
Dispersion Model ◽  
Small Diameter ◽  
Chemical Species ◽  
Taylor Dispersion ◽  
Phase Partitioning ◽  
Dispersion Mechanism ◽  
Effective Dispersion ◽  
Number Flow

An asymptotic analysis is presented for the advection–diffusion transport of a chemical species in flow through a small-diameter tube, where the flow consists of steady and oscillatory components, and the species may undergo linear reversible (phase exchange or wall retention) and irreversible (decay or absorption) reactions at the tube wall. Both developed and transient concentrations are considered in the analysis; the former is governed by the Taylor dispersion model, while the latter is required in order to formulate proper initial data for the developed mean concentration. The various components of the effective dispersion coefficient, valid when the developed state is attained, are derived as functions of the Schmidt number, flow oscillation frequency, phase partitioning and kinetics of the two reactions. Being more general than those available in the literature, this effective dispersion coefficient incorporates the combined effects of wall retention and absorption on the otherwise classical Taylor dispersion mechanism. It is found that if the phase exchange reaction kinetics is strong enough, the dispersion coefficient is probably to be increased by orders of magnitude by changing the tube wall from being non-retentive to being just weakly retentive.

scholarly journals Numerical Modeling of Contaminant Transport with Spatially-Dependent Dispersion and Non-Linear Chemical Reaction

2007 ◽  
Vol 12 (3) ◽  
pp. 329-343 ◽  
Author(s):  
A. J. Chamkha
Keyword(s):  
Contaminant Transport ◽  
Chemical Reaction ◽  
Numerical Solutions ◽  
Dispersion Coefficient ◽  
Transport Model ◽  
Physical Parameters ◽  
Contaminant Concentration ◽  
Dimensionless Time ◽  
Polynomial Type ◽  
Special Cases

A one-dimensional advective-dispersive contaminant transport model with scale-dependent dispersion coefficient in the presence of a nonlinear chemical reaction of arbitrary order is considered. Two types of variations of the dispersion coefficient with the downstream distance are considered. The first type assumes that the dispersivity increases as a polynomial function with distance while the other assumes an exponentiallyincreasing function. Since the general problem is nonlinear and possesses no analytical solutions, a numerical solution based on an efficient implicit iterative tri-diagonal finitedifference method is obtained. Comparisons with previously published analytical and numerical solutions for special cases of the main transport equation are performed and found to be in excellent agreement. A parametric study of all physical parameters is conducted and the results are presented graphically to illustrate interesting features of the solutions. It is found that the chemical reaction order and rate coefficient have significant effects on the contaminant concentration profiles. Furthermore, the scale-dependent polynomial type dispersion coefficient is predicted to obtain significant changes in the contaminant concentration at all dimensionless time stages compared with the constant dispersion case. However, relatively smaller changes in the concentration level are predicted for the exponentially-increasing dispersion coefficient.

scholarly journals DETERMINATION OF TRANSPORT PARAMETERS FOR SOLUTES IN SALT-TREATED SOIL COLUMNS

2017 ◽  
Vol 48 (1) ◽  
Author(s):  
Bahia & Naser
Keyword(s):  
Sodium Chloride ◽  
Calcium Chloride ◽  
Peclet Number ◽  
Dispersion Coefficient ◽  
Breakthrough Curves ◽  
Retardation Factor ◽  
Transport Parameters ◽  
Péclet Number ◽  
Mixed Salts ◽  
Sodium And Potassium

A laboratory experiment was carried out at the Department of Soil Sciences and Water Resources, College of Agriculture, University of Baghdad. Silty clay soil was treated with three salt solutions (NaCl, CaCl2 and mixed NaCl–CaCl2). Homogeneously packed soil columns (10 cm, 40 cm) were leached six times using tap water. Effluent samples were collected to determine ion concentration Cl-, Ca++, Na+, K+ and Mg++. Breakthrough curves were used to estimate solute transport parameters (retardation factor, peclet number) using an analytical solution of convection-dispersion equation (CDE) by CXTFIT program. The results showed that relative concentration of chloride was increased rapidly with calcium chloride, which increased sodium leaching rate at starting of breakthrough curve. Sodium chloride increased water requirements for calcium displacement. Results indicated a good fitting of convection-dispersion equation with breakthrough curves data. The best-fit were used to calculate peclet number, retardation factor and dispersion coefficient. When soil was treated with calcium chloride, Peclet number of chloride was increased from 3.13 to 6.48, while it has been decreased for calcium, sodium and potassium. Sodium chloride decreased peclet numbers of chloride, calcium and sodium. Also mixed salts increased sodium peclet number from 1.01 to 9.02. Results showed, calcium chloride decreased retardation factor of chloride from 1.59 to 0.50, while it has been increased from 1.39, 1.58 to 175.00, 493.36 for each of sodium and potassium, respectively. Retardation factor of calcium was decreased when soil was treated with sodium chloride or mixed salts. Dispersion coefficient was decreased for chloride, and increased for calcium and magnesium. When soil was treated with calcium chloride, dispersion coefficients have been increased from 24.29, 25.56 to 40.51, 40.89 cm2hr-1 for sodium and potassium, respectively.

scholarly journals Nusselt numbers for Poiseuille flow over isoflux parallel ridges accounting for meniscus curvature

2016 ◽  
Vol 811 ◽  
pp. 315-349 ◽  
Author(s):  
Toby L. Kirk ◽  
Marc Hodes ◽  
Demetrios T. Papageorgiou
Keyword(s):  
Nusselt Number ◽  
Poiseuille Flow ◽  
Numerical Solutions ◽  
Surface Deformation ◽  
Slip Length ◽  
Constant Heat Flux ◽  
Flux Boundary Condition ◽  
Nusselt Numbers ◽  
Dual Series Equations ◽  
Hydrodynamic Slip

We investigate forced convection in a parallel-plate-geometry microchannel with superhydrophobic walls consisting of a periodic array of ridges aligned parallel to the direction of a Poiseuille flow. In the dewetted (Cassie) state, the liquid contacts the channel walls only at the tips of the ridges, where we apply a constant-heat-flux boundary condition. The subsequent hydrodynamic and thermal problems within the liquid are then analysed accounting for curvature of the liquid–gas interface (meniscus) using boundary perturbation, assuming a small deflection from flat. The effects of this surface deformation on both the effective hydrodynamic slip length and the Nusselt number are computed analytically in the form of eigenfunction expansions, reducing the problem to a set of dual series equations for the expansion coefficients which must, in general, be solved numerically. The Nusselt number quantifies the convective heat transfer, the results for which are completely captured in a single figure, presented as a function of channel geometry at each order in the perturbation. Asymptotic solutions for channel heights large compared with the ridge period are compared with numerical solutions of the dual series equations. The asymptotic slip length expressions are shown to consist of only two terms, with all other terms exponentially small. As a result, these expressions are accurate even for heights as low as half the ridge period, and hence are useful for engineering applications.

Modelization of dispersion of swimming bacteria in Poiseuille flow

2021 ◽  
Author(s):  
Akash Ganesh ◽  
Romain Rescanieres ◽  
Carine Douarche ◽  
Harold Auradou
Keyword(s):  
Shear Rate ◽  
Numerical Simulations ◽  
Poiseuille Flow ◽  
Dispersion Coefficient ◽  
Equations Of Motion ◽  
Critical Shear Rate ◽  
Uniform Shear ◽  
Simple Boundary ◽  
Swimming Bacteria ◽  
Local Shear

<p>We study the shear-induced migration of dilute suspensions of swimming bacteria (modelled as Active elongated Brownian Particles or ABPs) subject to plane Poiseuille flow in a confined channel. By incorporating very simple boundary conditions, we perform numerical simulations of the 3D equations of motion describing the change in position and orientation of the particles. We investigate the effects of confinement, of non-uniform shear and of aspect ratio of the particles on the overall dynamics of the ABPs population.</p><p>We particularly study the coupling between the local shear and the change in the orientation of the particles. We thus perform numerical simulations on both the case where the change in the orientation of the ABPs is purely diffusive (decoupled case) and the case where their orientation is coupled to the shear flow (coupled case). We observe that the decoupled case exhibits a Taylor dispersion <em>i.e.</em>  the effective dispersion coefficient of the ABPs along the direction of the flow is proportional to the square of the imposed shear at all shears. </p><p>However, for all the coupled cases we observe a transition from a Taylor to an active-Taylor regime at a critical shear rate, indicating the effect of shear coupling on the orientation dynamics of the particles. This critical shear rate is directly correlated to the degree of confinement. The change in the dispersion coefficient along the direction of the flow as function of the shear rate is in qualitative agreement with previous studies[1]. </p><p>To further understand these results, we also investigate the change in the dispersion coefficient in the other two directions along with the effect of the shape of the particles. We believe that this study should enhance our understanding of dispersion of bacteria through porous media, on surfaces etc. where shear flows are ubiquitous. </p><p>[1] Sandeep Chilukuri, Cynthia H.Collins, and Patrick T. Underhill. Dispersionof flagellated swimming microorganisms in planar poiseuille flow.Physics offluids, 27, (031902):1 –17, 2015</p>

Pore-scale Mixing and the Evolution from Anomalous to Normal Hydrodynamic Dispersion

2021 ◽  
Author(s):  
Marco Dentz ◽  
Alexandre Puyguiraud ◽  
Philippe Gouze
Keyword(s):  
Porous Media ◽  
Large Scale ◽  
Molecular Diffusion ◽  
Pore Space ◽  
Breakthrough Curves ◽  
Heterogeneous Porous Media ◽  
Hydrodynamic Dispersion ◽  
Pore Scale ◽  
Sandstone Sample ◽  
Flow Variability

<p>Transport of dissolved substances through porous media is determined by the complexity of the pore space and diffusive mass transfer within and between pores. The interplay of diffusive pore-scale mixing and spatial flow variability are key for the understanding of transport and reaction phenomena in porous media. We study the interplay of pore-scale mixing and network-scale advection through heterogeneous porous media, and its role for the evolution and asymptotic behavior of hydrodynamic dispersion. In a Lagrangian framework, we identify three fundamental mechanisms of pore-scale mixing that determine large scale particle motion: (i) The smoothing of intra-pore velocity contrasts, (ii) the increase of the tortuosity of particle paths, and (iii) the setting of a maximum time for particle transitions. Based on these mechanisms, we derive an upscaled approach that predicts anomalous and normal hydrodynamic dispersion based on the characteristic pore length, Eulerian velocity distribution and Péclet number. The theoretical developments are supported and validated by direct numerical flow and transport simulations in a three-dimensional digitized Berea sandstone sample obtained using X-Ray microtomography. Solute breakthrough curves, are characterized by an intermediate power-law behavior and exponential cut-off, which reflect pore-scale velocity variability and intra-pore solute mixing. Similarly, dispersion evolves from molecular diffusion at early times to asymptotic hydrodynamics dispersion via an intermediate superdiffusive regime. The theory captures the full evolution form anomalous to normal transport behavior at different Péclet numbers as well as the Péclet-dependence of asymptotic dispersion. It sheds light on hydrodynamic dispersion behaviors as a consequence of the interaction between pore-scale mixing and Eulerian flow variability. </p>

scholarly journals The Hydrodynamic Dispersion Characteristics of Coral Sands

2019 ◽  
Vol 7 (9) ◽  
pp. 291 ◽  
Author(s):  
Xiang Cui ◽  
Changqi Zhu ◽  
Mingjian Hu ◽  
Xinzhi Wang ◽  
Haifeng Liu
Keyword(s):  
Particle Size ◽  
Porous Media ◽  
Flow Velocity ◽  
Molecular Diffusion ◽  
Dispersion Coefficient ◽  
Hydrodynamic Dispersion ◽  
Dispersion Characteristics ◽  
Factors Affecting ◽  
Mechanical Dispersion ◽  
Freshwater Aquifers

Dispersion characteristics are important factors affecting groundwater solute transport in porous media. In marine environments, solute dispersion leads to the formation of freshwater aquifers under islands. In this study, a series of model tests were designed to explore the relationship between the dispersion characteristics of solute in calcareous sands and the particle size, degree of compactness, and gradation of porous media, with a discussion of the types of dispersion mechanisms in coral sands. It was found that the particle size of coral sands was an important parameter affecting the dispersion coefficient, with the dispersion coefficient increasing with particle size. Gradation was also an important factor affecting the dispersion coefficient of coral sands, with the dispersion coefficient increasing with increasing d10. The dispersion coefficient of coral sands decreased approximately linearly with increasing compactness. The rate of decrease was −0.7244 for single-grained coral sands of particle size 0.25–0.5 mm. When the solute concentrations and particle sizes increased, the limiting concentration gradients at equilibrium decreased. In this study, based on the relative weights of molecular diffusion versus mechanical dispersion under different flow velocity conditions, the dispersion mechanisms were classified into five types, and for each type, a corresponding flow velocity limit was derived.

scholarly journals Removal of acetylsalicylic acid (ASA) in packed microcolumns with carbon xerogel modified with TiO2 nanoparticles

2019 ◽  
Vol 39 (2) ◽  
Author(s):  
Viviana Eloisa Gomez Rengifo ◽  
Adriana Herrera Barros ◽  
Jorge Hernan Sanchez Toro
Keyword(s):  
Acetylsalicylic Acid ◽  
Adsorption Capacity ◽  
Tio2 Nanoparticles ◽  
Dispersion Model ◽  
Packed Bed ◽  
Breakthrough Curves ◽  
Bet Surface Area ◽  
Maximum Adsorption Capacity ◽  
Carbon Xerogel ◽  
Complementary Techniques

The adsorption capacity of acetylsalicylic acid was evaluated using carbon xerogel (CX) and carbon xerogel modified with TiO2 nanoparticles (CXM). These materials were characterized by different techniques such as Scanning Electron Microscopy (SEM), X-Ray Diffraction (XRD), and Fourier Transform Infrared (FTIR) spectroscopy. BET surface area measurements found values of 762 m2/g and 214 m2/g for CX and CXM, respectively. Batch experiments show that the Langmuir-Freundlich model best represents the experimental adsorption isotherm, in addition to show a maximum adsorption capacity of 17,48 mg/g.  In continuous experiments, the effect of the inlet concentration and flow rate on the adsorption capacity of the micro-packed bed adsorber were evaluated. Breakthrough curves agree well with the axial dispersion model. In view of their adsorption capacity, carbon xerogels provide a potential material for the removal of emergent contaminants from the pharmaceutical industry. Besides, the incorporation of TiO2 nanoparticles allows the implementation of complementary techniques, e.g. photodegradation, as an alternative to achieve higher elimination of aqueous contaminants.

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