Skip to main content
SpringerLink
Log in

A Survey of Multicomponent Mass Diffusion Flux Closures for Porous Pellets: Mass and Molar Forms

  • Published:
Transport in Porous Media Aims and scope Submit manuscript

Abstract

Heterogeneous catalysis is of paramount importance in many areas of gas conversion and processing in chemical engineering industries. In porous pellets, the catalytic reactions may be affected by diffusional limitations such that the global rate can be different from the intrinsic reaction rate. In the literature, a number of multicomponent diffusion flux closures have been applied to characterize the diffusion process within different units in chemical process plants. The main purpose of this paper is to outline the derivation of the different diffusion flux models: the rigorous Maxwell–Stefan and dusty gas models, and the simpler Wilke and Wilke–Bosanquet models. Usually the diffusion fluxes are derived and presented with respect to the molar average velocity definition. In this study, also the diffusion flux closures with respect to the mass average velocity definition is outlined. Thus, if the temperature equation and the momentum equation are used in the pellet model, a consistently closed set of pellet equations is obtained on mass basis holding only the mass average velocity. On the other hand, for the closed set of pellet equations on molar basis, the component balances hold the molar averaged velocity whereas the temperature and momentum equations hold the mass average velocity due to the physical laws applied deriving these fundamental balances. Nevertheless, the Maxwell–Stefan and dusty gas models are manipulated and put on the convenient Fickian form. The second purpose of this article is the evaluation of the diffusion flux closures derived. For this purpose, a transient model is developed to describe the evolution of the species composition, pressure, velocity, temperature, total concentration, and fluxes within a spherical pellet. The catalyst problem has been simulated for the methanol dehydration process producing dimethyl ether (DME), with computed efficiency factor values in the range 0.06–0.6 for pellet pore diameters of 0.1–100 nm. Identical results are expected for the mole and mass based pellet equations. However, deviations are obtained in the component fractions comparing the mass and mole based pellet model formulations where the mass fluxes were described according to the Wilke and Wilke–Bosanquet models. On the other hand, the rigorous Maxwell–Stefan and dusty gas models gave identical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

B :

Permeability, m2

c :

Concentration, kmol/m3

Cp :

Heat capacity, J/(kmol K)

Cp′′:

Heat capacity, J/(kg K)

D :

Diffusivity, m2/s

\({\widetilde D_{ij}}\) :

Maxwell–Stefan diffusivities, m2/s

D ij :

Binary diffusion coefficient for species i and j, m2/s

D im :

Diffusion coefficient of species i in Wilke molar flux, m2/s

\({D_{i{\rm m}}^{''}}\) :

Diffusion coefficient of species i in Wilke mass flux, m2/s

D iK :

Knudsen diffusion coefficient for species i, m2/s

D i,eff :

Bosanquet diffusivity, mole basis, m2/s

\({D''_{i,{\rm eff}}}\) :

Bosanquet diffusivity, mass basis, m2/s

d i :

Diffusional driving force, m−1

d p :

Diameter of pellet, m

d pore :

Average pore diameter, m

g :

External force per unit mass, m/s2

h :

Heat transfer coefficient, W/(m2K)

ΔH :

Heat of reaction, J/kmol

J :

Molecular diffusion flux, kmol/(s m2)

j :

Mass diffusion flux, kg/(s m2)

k :

Reaction rate constant, kmol/(kg h)

k :

Boltzmann constant, 1.3805 · 10−23, J/K

k :

Mass transfer coefficient, m/s

K M, K W :

Adsorption constant, m3/kmol

K :

Thermodynamic equilibrium constant, dimensionless

M :

Molecular weight, kg/kmol

N :

Number of collocation points, dimensionless

n :

Number of species in gas mixture, dimensionless

n i :

Combined mass flux of species i in gas mixture, kg/(m2s)

N i :

Combined molar flux of species i in gas mixture, kmol/(m2s)

p :

Pressure, Pa

Q :

Heat conductivity flux, J/(m2s)

R :

Gas constant, J/(kmol K)

r :

Radial coordinate, m

r :

Reaction rate, kmol/(kg s)

r M :

Reaction rate, kmol/(kg h)

r p :

Radius of pellet, m

S :

Molecular source term, (kmol)/(m3s)

S′′:

Mass source term, kg/(m3s)

Ŝ :

Heat source term, J/(m3s)

T :

Temperature, K

t :

Time, s

u :

Molar average velocity, m/s

v :

Mass average velocity, m/s

v i :

Velocity of species i with respect to fixed coordinates, m/s

V :

Volume, m3

x :

Mole fraction, dimensionless

η :

Effectiveness factor, dimensionless

\({\epsilon}\) :

Porosity, dimensionless

\({\epsilon}\) :

Characteristic Lennar-Jones energy, J

λ :

Conductivity, W/(m K)

ρ :

Density, kg/m3

τ :

Tortuosity, dimensionless

μ :

Dynamic viscosity, kg/(m s)

ω :

Mass fraction, dimensionless

Ω :

Diffusion collision integral, dimensionless

Σ :

Characteristic Lennard-Jones length, Å

b:

Bulk

i, j, n :

Species type

M:

Methanol

p:

Pellet

r :

Radial direction

ref:

Reference state

W:

Water

*:

Dimensionless variable

′:

Gas mixture with dust particles

′′:

Mass basis

′′:

Second-order derivative in finite difference scheme (69)

b:

Bulk

e:

Effective

s:

Superficial

s:

Pellet surface; equation (71)

τ :

Time level

References

  • Berčič G., Levec J.: Intrinsic and global reaction rate of methanol dehydration over γ−Al2O3 pellets. Ind. Eng. Chem. Res. 31, 1035–1040 (1992)

    Article  Google Scholar 

  • Berčič G., Levec J.: Catalytic dehydration of methanol to dimethyl ether. Kinetic investigation and reactor simulation. Ind. Eng. Chem. Res. 32, 2478–2484 (1993)

    Article  Google Scholar 

  • Bhatia S.K., Nicholson D.: Some pitfalls in the use of the Knudsen equation in modelling diffusion in nanoporous materials. Chem. Eng. Sci. 66, 284–293 (2011)

    Article  Google Scholar 

  • Bird R.B., Stewart W.E., Lightfoot E.N.: Transport Phenomena. Wiley, New York (2002)

    Google Scholar 

  • Bosanquet, C. H.: Wall effects in gas-temperature measurements. British TA Report BR-507 (1944)

  • Burghardt A., Aerts J.: Pressure changes during diffusion with chemical reaction in a porous pellet. Chem. Eng. Process. 23, 77–87 (1988)

    Article  Google Scholar 

  • Curtiss C.F., Hirschfelder J.O.: Transport properties of multicomponent gas mixtures. J. Chem. Phys. 17, 550–555 (1949)

    Article  Google Scholar 

  • Datta R., Vilekar S.A.: The continuum mechanical theory of multicomponent diffusion in fluid mixtures. Chem. Eng. Sci. 65, 5976–5989 (2010)

    Article  Google Scholar 

  • Diep B.T.: Thermodynamic equilibrium constants for the methanol–dimethyl ether–water system. J. Chem. Eng. Data 32, 330–333 (1987)

    Article  Google Scholar 

  • Dogˇu T.: Diffusion and reaction in catalyst pellets with bidisperse pore size distribution. Ind. Eng. Chem. Res. 37, 2158–2171 (1998)

    Article  Google Scholar 

  • Elnashaie S.S.E.H., Abashar M.E.E.: Steam reforming and methanation effectiveness factors using the dusty gas model under industrial conditions. Chem. Eng. Process. 32, 177–189 (1992)

    Google Scholar 

  • Fick A.: On liquid diffusion. Philos. Mag. J. Sci. 10, 31–39 (1855)

    Google Scholar 

  • Finlayson B.A.: The Method of Weighted Residuals and Variational Principles: Whit Applications in Fluid Mechanics, Heat and Mass Transfer, vol. 87 of Mathematics in Science and Engineering. Academic Press, New York (1972)

    Google Scholar 

  • Geankoplis C.J.: Transport processes and separation process principles. Prentice Hall, Upper Saddle River (2003)

    Google Scholar 

  • Graaf G.H., scholtens H., Stamhuis E.J., Beenackers A.A.C.M.: Intra-particle diffusion limitations in low-pressure methanol synthesis. Chem. Eng. Sci. 45, 773–783 (1990)

    Article  Google Scholar 

  • Horák Z., Schneider P.: Comparison of some models of porous media for gas diffusion. Chem. Eng. J. 2, 26–36 (1971)

    Article  Google Scholar 

  • Jackson R.: Transport in Porous Catalysts. Elsevier, Amsterdam (1977)

    Google Scholar 

  • Jakobsen H.A.: Chemical Reactor Modeling. Multiphase Reactive Flows. Springer, Berlin (2008)

    Google Scholar 

  • Jakobtorweihen S., Lowe C.P., Keil F.J., Smit B.: Diffusion of chain molecules and mixtures in carbon nanotubes: The effect of host lattice flexibility and theory of diffusion in the Knudsen regime. J. Chem. Phys. 127, 024904 (2007)

    Article  Google Scholar 

  • Kleijn, C.R.: Transport phenomena in chemical vapor deposition reactors. Ph.D. thesis, Technische Universiteit, Delft (1991)

  • Knudsen M.: The laws of the molecular current and the internal friction current of gases by channels. Ann. Phys. (Leipzig) 29, 75–130 (1909)

    Google Scholar 

  • Knudsen M., Fisher W.J.: The molecular and the frictional flow of gases in tubes. Phys. Rev. 31, 586–588 (1910)

    Google Scholar 

  • Krishna R.: Problems and pitfalls in the use of the Fick formulation for intraparticle diffusion. Chem. Eng. Sci. 48(5), 845–961 (1993)

    Article  Google Scholar 

  • Krishna R.: Describing the diffusion of guest molecules inside porous structures. J. Phys. Chem. C 113, 19756–19781 (2009)

    Article  Google Scholar 

  • Krishna R., van Baten J.M.: Influence of adsorption on the diffusion selectivity for mixture permeation across mesoporous membranes. J. Membr. Sci. 369, 545–549 (2011a)

    Article  Google Scholar 

  • Krishna R., van Baten J.M.: Investigating the validity of the Knudsen prescription for diffusivities in a mesoporous covalent organic framework. Ind. Eng. Chem. Res. 50, 7083–7087 (2011b)

    Article  Google Scholar 

  • Krishna R., van Baten J.M.: A molecular dynamics investigation of the unusual concentration dependencies of Fick diffusivities in silica mesopores. Microporous Mesoporous Mater. 138, 228–234 (2011c)

    Article  Google Scholar 

  • Krishna R., van Baten J.M.: Investigating the validity of the bosanquet formula for estimation of diffusivites in mesopores. Chem. Eng. Sci. 69, 684–688 (2012)

    Article  Google Scholar 

  • Krishna R., Wesselingh J.A.: The Maxwell–Stefan approach to mass transfer. Chem. Eng. Sci. 52(6), 861–911 (1996)

    Google Scholar 

  • Kuijlaars K.J., Kleijn C.R., van den Akker H.E.A.: Multi-component diffusion phenomena in multiple-wafer chemical vapour deposition reactors. Chem. Eng. J. 57, 127–136 (1995)

    Google Scholar 

  • Lommerts B.J., Graaf G.H., Beenackers A.: Mathematical modeling of internal mass transport limitations in methanol synthesis. Chem. Eng. Sci. 55, 5589–5598 (2000)

    Article  Google Scholar 

  • Lovik I., Hillestad M., Hertzberg T.: Long term dynamic optimization of a catalytic reactor system. Comput. Chem. Eng. 22, S707–S710 (1998)

    Article  Google Scholar 

  • Mason E.A., Malinauskas A.P.: Gas Transport in Porous Media: The Dusty Gas Model. Elsevier, Amsterdam (1983)

    Google Scholar 

  • Maxwell J.C.: On the dynamical theory of gases. Phil. Trans. R. Soc. 157, 49–88 (1866)

    Article  Google Scholar 

  • Reid R.C., Prausnitz J.M., Poling B.E.: The properties of gases and liquids. McGraw-Hill, New York (1987)

    Google Scholar 

  • Rezaie N., Jahanmiri A., Moghtaderi B., Rahimpour M.R.: A comparison of homogeneous and heterogeneous dynamic models for industrial methanol reactors in the presence of catalyst deactivation. Chem. Eng. Process. 44, 911–921 (2005)

    Article  Google Scholar 

  • Rice R.G., Do D.D.: Applied mathematics and modeling for chemical engineers. Wiley, New York (1995)

    Google Scholar 

  • Rout K.R., Solsvik J., Nayak A.K., Jakobsen H.A.: A numerical study of multicomponent mass diffusion and convection in porous pellets for the sorption-enhanced steam methane reforming and desorption processes. Chem. Eng. Sci. 66, 4111–4126 (2011)

    Article  Google Scholar 

  • Sahimi M., Gavalas G.R., Tsotsis T.T.: Statistical and continuum models of fluid–solid reactions in porous media. Chem. Eng. Sci. 45, 1443–1502 (1990)

    Article  Google Scholar 

  • Salmi T., Wärnå J.: Modelling of catalytic packed-bed reactors—comparison of different diffusion models. Comput. Chem. Eng. 15, 715–727 (1991)

    Article  Google Scholar 

  • Serrin J.: Mathematical principles of classical fluid mechanics. In: Flugge, S., Truesdell, C. (eds) Handbuch der physik, vol. VIII, Springer, New York (1959)

    Google Scholar 

  • Shahrokhi M., Baghmisheh G.R.: Modeling, simulation and control of a methanol synthesis fixed-bed reactor. Chem. Eng. Sci. 60, 4275–4286 (2005)

    Article  Google Scholar 

  • Smith J.M.: Chemical engineering kinetics, 3rd edn. McGraw-Hill, Singapore (1981)

    Google Scholar 

  • Solsvik J., Jakobsen H.A.: A numerical study of a two property catalyst/sorbent pellet design for the sorption-enhanced steam–methane reforming process: modeling complexity and parameter sensitivity study. Chem. Eng. J. 178, 407–422 (2011)

    Article  Google Scholar 

  • Solsvik, J., Jakobsen, H.A.: Multicomponent mass diffusion in porous pellets: Effects of flux models on the pellet level and impacts on the reactor level. Application to methanol synthesis. Can. J. Chem. Eng. doi:10.1002/cjce.20680(2012)

  • Stefan, J.: Über das Gleichgewicht und die Bewegung insbesondere die Diffusion von Gasgemengen. Sitzber. Akad. Wiss. Wien 63, 63–124 (1871)

    Google Scholar 

  • Taylor R., Krishna R.: Multicomponent Mass Transfer. Wiley, New York (1993)

    Google Scholar 

  • Tomadakis M.M., Sotirchos S.V.: Ordinary, transition, and Knudsen regime diffusion in random capillary structures. Chem. Eng. Sci. 48(19), 3323–3333 (1993)

    Article  Google Scholar 

  • Vakili R., Pourazadi E., Eslamloueyan R., Rahimpour M.R.: Direct dimethyl ether (DME) synthesis through a thermally coupled heat exchanger reactor. Appl. Energy 88, 1211–1223 (2011)

    Article  Google Scholar 

  • Wilke C.R.: Diffusional properties of multicomponent gases. Chem. Eng. Prog. 46, 95–104 (1950)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), 7491, Trondheim, Norway

    Jannike Solsvik & Hugo A. Jakobsen

Authors
  1. Jannike Solsvik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hugo A. Jakobsen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jannike Solsvik.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Solsvik, J., Jakobsen, H.A. A Survey of Multicomponent Mass Diffusion Flux Closures for Porous Pellets: Mass and Molar Forms. Transp Porous Med 93, 99–126 (2012). https://doi.org/10.1007/s11242-012-9946-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11242-012-9946-7

Keywords

Search

Navigation