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Mass transfer and dispersion around an active cylinder in cross flow and buried in a packed bed

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Abstract

The present work describes the mass transfer process between a moving fluid and a slightly soluble cylinder, with the axis perpendicular to flow direction, buried in a packed bed of small inert particles, with uniform voidage. Fluid flow in the packed bed around the cylinder was assumed to follow Darcy’s law and, at each point, dispersion of solute was assumed to be determined by radial and axial dispersion coefficients, in the cross-stream and streamwise directions, respectively. Numerical solutions of the differential equation describing solute mass conservation were undertaken to obtain the concentration field near the soluble surface and the mass transfer flux was integrated to give the Sherwood number as a function of the relevant parameters. Mathematical expressions are proposed that describes accurately the dependence found numerically between the value of the Sherwood number and the values of Peclet number, Schmidt number and the ratio between the diameter of cylinder and the diameter of inerts.

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Abbreviations

A :

dimensionless parameter (defined in Eq. 8a)

B :

dimensionless parameter (defined in Eq. 8b)

c :

solute concentration

c 0 :

bulk concentration of solute

c*:

equilibrium concentration of solute

C :

dimensionless solute concentration

d :

diameter of inert particles

d c :

diameter of soluble cylinder

D L :

longitudinal (axial) dispersion coefficient

D m :

molecular diffusion coefficient

D′m :

effective molecular diffusion coefficient (=D m/τ)

D T :

transverse (radial) dispersion coefficient

K :

permeability in Darcy’s law

k :

mass transfer coefficient

L :

length of soluble cylinder

n :

mass transfer rate

P :

pressure

Pe L (∞):

asymptotic value of Pe L when Re p → ∞

Pe T (∞):

asymptotic value of Pe T when Re p → ∞

R :

radius of soluble cylinder

ℜ:

dimensionless spherical radial co-ordinate

r :

spherical radial coordinate (distance to the centre of the soluble cylinder)

u :

absolute value of interstitial velocity

U :

interstitial velocity (vector)

u 0 :

absolute value of interstitial velocity far way from cylinder

u r , u θ :

components of fluid interstitial velocity

ɛ:

bed voidage

Φ:

dimensionless potential function

ϕ:

potential function

η:

enhancement factor due to convective dispersion

μ:

dynamic viscosity

θ:

spherical angular coordinate

ρ:

density

τ:

tortuosity

Ψ:

dimensionless stream function

ψ:

stream function

Pe′:

Peclet number based on diameter of soluble cylinder (=u 0 d c/D′m)

Pe′p :

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

Re p :

Reynolds number (=ρ u 0 d/μ)

Sc :

Schmidt number (=μ /ρ D m)

Sh′:

Sherwood number (=kd c/D′m)

Sh′md :

Sherwood number when D T≅ D L ≅ D′m (i.e. Pe′p< 0.2)

i, j :

grid node indices (see Fig. 2)

n :

iteration number

References

  • Alves MA, Oliveira PJ, Pinho FT (2003) A convergent and universally bounded interpolation scheme for the treatment of advection. Int J Numer Meth Fluids 41:47–75

    Article  MATH  Google Scholar 

  • Anderson JD (1995) Computational fluid dynamics. McGraw Hill, New York

    Google Scholar 

  • Bear J (1972) Dynamics of fluids in porous media. Elsevier, New York

    Google Scholar 

  • Burmester SSH, Coelho MAN, Guedes de Carvalho JRF (1990) Transverse dispersion in granular beds: part IV—Mass transfer around an active cylinder with the axis perpendicular to the flow direction in a packed bed. Trans IChemE 68:510–515

    Google Scholar 

  • Coelho MAN, Guedes de Carvalho JRF (1988a) Transverse dispersion in granular beds: part I—Mass transfer from a wall and the dispersion coefficient in packed beds. Chem Eng Res Des 66:165–177

    Google Scholar 

  • Coelho MAN, Guedes de Carvalho JRF (1988b) Transverse dispersion in granular beds: part II—Mass-transfer from large spheres immersed in fixed or fluidized-beds of small inert particles. Chem Eng Res Des 66:178–189

    Google Scholar 

  • Currie IG (1993) Fundamental mechanics of fluids. McGraw Hill, New York

    Google Scholar 

  • Darwish MS, Moukalled F (1994) Normalized variable and space formulation methodology for high-resolution schemes. Numer Heat Tranf B Fundam 26:79–96

    Article  Google Scholar 

  • Ferziger JH, Peric M (1996) Computational methods for fluid dynamics. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  • Gaskell PH, Lau AKC (1988) Curvature compensated convective transport: SMART, a new boundedness preserving transport algorithm. Int J Numer Methods Fluids 8:617–641

    Article  MATH  MathSciNet  Google Scholar 

  • Guedes de Carvalho JRF, Delgado JMPQ (2005) Overall map and correlation of dispersion data for flow through granular packed beds. Chem Eng Sci 60:365–375

    Article  Google Scholar 

  • Guedes de Carvalho JRF, Delgado JMPQ, Alves MA (2004) Mass transfer between flowing fluid and sphere buried in packed bed of inerts. AIChE J 50:65–74

    Article  Google Scholar 

  • Gunn DJ (1968) Mixing in packed and fluidised beds. Chem Engr Lond, CE153–172

  • Leonard BP (1988) Simple high-accuracy resolution program for convective modeling of discontinuities. Int J Numer Methods Fluids 8:1291–1318

    Article  MATH  Google Scholar 

  • Scheidegger AE (1974) The physics of flow through porous media, 3rd edn. University of Toronto Press, Toronto

    Google Scholar 

  • Wilhelm RH (1962) Progress towards the a priori design of chemical reactors. Pure Appl Chem 5:403–421

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Departamento de Engenharia Química, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 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. Mass transfer and dispersion around an active cylinder in cross flow and buried in a packed bed. Heat Mass Transfer 42, 1119–1128 (2006). https://doi.org/10.1007/s00231-005-0076-4

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  • DOI: https://doi.org/10.1007/s00231-005-0076-4

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