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Concentration distribution in the wake of a sphere buried in a granular bed through which fluid flows

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Abstract

The concentration distribution in the wake of a soluble sphere immersed in a granular bed of inert particles, through which fluid flows with “uniform velocity”, has been obtained numerically, for solute transport by both advection and diffusion/dispersion. Fluid flow in the granular bed around the sphere was assumed to follow Darcy’s law and, at each point, dispersion of solute was considered in both the cross-stream and streamwise directions. The elliptic PDE equation, resulting from a differential material balance on the solute, was solved numerically over a wide range of values of the relevant parameters (Peclet number and Schmidt number). The solution gives the concentration contour plots and, for each concentration level, the width and downstream length of the corresponding contour surface were determined. General expressions are presented to predict contaminant “plume” size downstream of the polluting source.

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Abbreviations

a :

radius of the active sphere (m)

c :

solute concentration (kg/m3)

c 0 :

bulk solute concentration (kg/m3)

c * :

saturation solute concentration (kg/m3)

C :

dimensionless solute concentration (−)

d :

diameter of inert particles (m)

d 1 :

diameter of active sphere (m)

D L :

longitudinal dispersion coefficient (m2/s)

D m :

molecular diffusion coefficient (m2/s)

\(\hbox{\it D}^{\prime}_{\rm m}\) :

effective molecular diffusion coefficient (=D m/τ) (m2/s)

D T :

transverse (radial) dispersion coefficient (m2/s)

K :

permeability in Darcy’s law (m3 s/kg)

p :

pressure (kg/ms2)

Pe L(∞):

asymptotic value of Pe L when Re p → ∞ (−)

Pe T(∞):

asymptotic value of Pe T when Re p → ∞ (−)

ℜ:

dimensionless spherical radial co-ordinate (= r/a) (−)

r :

spherical radial coordinate (distance to the centre of the soluble sphere) (m)

U :

dimensionless interstitial velocity (= u/u 0) (−)

u :

magnitude of interstitial velocity (m/s)

u :

interstitial velocity (vector) (m/s)

u 0 :

magnitude of interstitial velocity far from the active sphere (m/s)

u r ,u θ :

components of fluid interstitial velocity vector (m/s)

x :

streamwise co-ordinate (m)

x C /a :

x-axis dimensionless wake size of the C concentration front (−)

(x C /a)md :

x C /a value when D T = D L = \(\hbox{\it D}^{\prime}_{\rm m}\) (i.e. \(\hbox{\it Pe}^{\prime}_{\rm p}\) < 0.1) (−)

y :

cross-streamwise co-ordinate (m)

y C/a :

y-axis dimensionless wake size of the C concentration front (−)

(y C /a)md :

y C /a value when D T = \(\hbox{\it D}^{\prime}_{\rm m}\) (i.e. \(\hbox{\it Pe}^{\prime}_{\rm p}\) < 0.1) (−)

ɛ:

bed voidage (−)

Φ:

dimensionless potential function (−)

ϕ:

potential function (defined in Eq. 1) (m2/s)

η:

corrective factor due to convective dispersion (−)

μ:

dynamic viscosity (kg/ms)

θ:

spherical angular coordinate (rad)

ρ:

density (kg/m3)

τ:

tortuosity (−)

Ψ:

dimensionless stream function (−)

ψ:

stream function (defined in Eq. 2) (m3/s)

Pe′:

Peclet number based on diameter of active sphere (=u 0 d 1/D ′m ) (−)

Pe ′p :

Peclet number based on diameter of inert particles (=u 0 d/D ′m ) (−)

Re p :

Reynolds number based on diameter of inert particles (=ρUd/μ) (−)

Sc :

Schmidt number (=μ/ρD m) (−)

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Acknowledgment

The author wishes to thank Fundação para a Ciência e a Tecnologia for the Grant No SFRH/BPD/11639/2002.

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Authors and Affiliations

  1. Laboratório de Físicadas Construções Departamento de Engenharia Civil (LFC), Faculdade de Engenharia da, Universidade do Porto, Rua Dr. Roberto Frias, s/n, 4200-465, Porto, Portugal

    J. M. P. Q. Delgado

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  1. J. M. P. Q. Delgado
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Correspondence to J. M. P. Q. Delgado.

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Delgado, J.M.P.Q. Concentration distribution in the wake of a sphere buried in a granular bed through which fluid flows. Heat Mass Transfer 44, 1427–1434 (2008). https://doi.org/10.1007/s00231-008-0385-5

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