Multi-scale mathematical model of mass transference phenomena inside monolithic carbon aerogels

  • D. Camargo-TrillosEmail author
  • F. Chejne
  • J. Alean


A phenomenological basis model was developed to describe behavior of gas adsorption at multi-length scales; from the macroscale (fixed bed scale) to mass transport, into the mesopores and micropores (microscale). The multiscale mass transport model is based on partial differential equations of adsorbate in the gas phase; where an additional adsorption flux on interface was implemented as a boundary condition (BC). Therefore, parallel contributions of kinetic adsorption and diffusive mass transference at BC were considered. The model allows a good fit between experimental and simulated results for fixed bed (FB) concentration profile, height of mass transport, and total adsorption capacity by carbon aerogels, with mesopores to micropores volume relation from 0.3 to 3.4. Both the experimental setup date and multi-scale model identify volume relation (Vmeso/Vmicro) as a key parameter on the design and optimization of adsorption technologies.



Gas concentration, mmol m−3


Diffusion, m2 s−1


Henry constant by adsorption


Height Mass Transfer Zone, cm


Henry adsorption constant, mol/g


Micropore kinetic adsorption, s−1


Transfer external constant, cm s−1


Knudsen number


Mass flux, mmol m−2 s−1


Schmidt number


Sherwood number


Adsorption capacity, mmol/g-MCA


Radial axis


Characteristic monolithic particle radio, Cm


Specific surface area, m2 g−1


Temperature, K


Microparticle density, g cm−3


Time, Min


Volume, cm3


Interstitial velocity, m/s


Total standard micropore volume, cm3 g−1


Fractional capacity


Boundary layer thickness


Kinetic diameter













Mass Transfer Zone



Authors are thankful with the COLCIENCIAS (Ph.D. scholarship program number 528 and 617) for providing funding, for the successful completion of this study. Diego Camargo is grateful with CIDI-Universidad Pontifícia Bolivariana for supporting this work. Farid Chejne and Jader Alean wish to thank to the project “Strategy of transformation of the Colombian energy sector in the horizon 2030” funded by the call 788 of Colciencias Scientific Ecosystem. Contract number FP44842-210-2018.

Compliance with ethical standards

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.


  1. 1.
    Mao Z, Sinnott SB (2000) A computational study of molecular diffusion and dynamic flow through carbon nanotubes. J Phys Chem B 104(19):4618–4624CrossRefGoogle Scholar
  2. 2.
    Kuznetsova A et al (2001) Optimization of Xe adsorption kinetics in single walled carbon nanotubes. J Chem Phys 115(14):6691–6698CrossRefGoogle Scholar
  3. 3.
    Pekala RW et al (1992) Aerogels derived from multifunctional organic monomers. J Non-Cryst Solids 145:90–98CrossRefGoogle Scholar
  4. 4.
    Pierre AC, Pajonk GM (2002) Chemistry of aerogels and their applications. Chem Rev 102(11):4243–4266CrossRefGoogle Scholar
  5. 5.
    Kakaç S, Pramuanjaroenkij A (2009) Review of convective heat transfer enhancement with nanofluids. Int J Heat Mass Transf 52(13):3187–3196CrossRefzbMATHGoogle Scholar
  6. 6.
    Keblinski P et al (2002) Mechanisms of heat flow in suspensions of nano-sized particles (nanofluids). Int J Heat Mass Transf 45(4):855–863CrossRefzbMATHGoogle Scholar
  7. 7.
    Karger J, Valiullin R (2013) Mass transfer in mesoporous materials: the benefit of microscopic diffusion measurement. Chem Soc Rev 42(9):4172–4197CrossRefGoogle Scholar
  8. 8.
    Camargo-Trillos D, Chejne F, Pabón E, Carrasco-Marin F (2015) Effect on mass transference phenomena by textural change inside monolithic carbon aerogels. Heat Mass Transf 51(8):1141–1148CrossRefGoogle Scholar
  9. 9.
    Shahtalebi A et al (2014) Slow diffusion of methane in ultra-micropores of silicon carbide-derived carbon. Carbon 77:560–576CrossRefGoogle Scholar
  10. 10.
    Maldonado-Hódar FJ et al (2007) Reversible toluene adsorption on monolithic carbon aerogels. J Hazard Mater 148(3):548–552CrossRefGoogle Scholar
  11. 11.
    Lin C, Ritter JA (2000) Carbonization and activation of sol-gel derived carbon xerogels. Carbon 38(6):849–861CrossRefGoogle Scholar
  12. 12.
    Malek A, Farooq S (1997) Kinetics of hydrocarbon adsorption on activated carbon and silica gel. AICHE J 43(3):761–776CrossRefGoogle Scholar
  13. 13.
    Bonjour J, Chalfen J-B, Meunier F (2002) Temperature swing adsorption process with indirect cooling and heating. Ind Eng Chem Res 41(23):5802–5811CrossRefGoogle Scholar
  14. 14.
    Patton A, Crittenden BD, Perera SP (2004) Use of the linear driving force approximation to guide the Design of Monolithic Adsorbents. Chem Eng Res Des 82(8):999–1009CrossRefGoogle Scholar
  15. 15.
    Brosillon S, Manero M-H, Foussard J-N (2001) Mass transfer in VOC adsorption on zeolite: experimental and theoretical breakthrough curves. Environ Sci Technol 35(17):3571–3575CrossRefGoogle Scholar
  16. 16.
    Khajuria H, Pistikopoulos EN (2011) Dynamic modeling and explicit/multi-parametric MPC control of pressure swing adsorption systems. J Process Control 21(1):151–163CrossRefGoogle Scholar
  17. 17.
    Akinlabi CO et al. (2007) Modelling, design and optimisation of a hybrid PSA-membrane gas separation process, in Computer Aided Chemical Engineering, P. Valentin and A. Paul Şerban, Editors. Elsevier: 363–370Google Scholar
  18. 18.
    Wang J et al (2017) Experimental investigation of gas mass transport and diffusion coefficients in porous media with nanopores. Int J Heat Mass Transf 115:566–579CrossRefGoogle Scholar
  19. 19.
    Fletcher AJ, Yüzak Y, Thomas KM (2006) Adsorption and desorption kinetics for hydrophilic and hydrophobic vapors on activated carbon. Carbon 44(5):989–1004CrossRefGoogle Scholar
  20. 20.
    Ding LP, Bhatia SK, Liu F (2002) Kinetics of adsorption on activated carbon: application of heterogeneous vacancy solution theory. Chem Eng Sci 57(18):3909–3928CrossRefGoogle Scholar
  21. 21.
    Ding LP et al (2005) Heterogeneous model for gas transport in carbon molecular sieves. Langmuir 21(2):674–681CrossRefGoogle Scholar
  22. 22.
    Shahtalebi A et al (2014) Slow diffusion of methane in ultra-micropores of silicon carbide-derived carbon. Carbon 77(0):560–576CrossRefGoogle Scholar
  23. 23.
    Beers K (2007) Numerical methods for chemical engineering. Cambridge University PressGoogle Scholar
  24. 24.
    Vurro M, Castellano L Numerical treatments of the interface discontinuity in solid-water mass transfer problems. Comput Math Applics 45(4–5):785–788Google Scholar
  25. 25.
    Illingworth TC, Golosnoy IO (2005) Numerical solutions of diffusion-controlled moving boundary problems which conserve solute. J Comput Phys 209(1):207–225CrossRefzbMATHGoogle Scholar
  26. 26.
    Fairen Jimenez D (2006) Aerogeles Monolíticos de carbón como adsorbentes para la eliminación de compuestos orgánicos volátiles (BTX). Universidad de Granada, GrandaGoogle Scholar
  27. 27.
    Neimark AV et al (2009) Quenched solid density functional theory and pore size analysis of micro-mesoporous carbons. Carbon 47(7):1617–1628CrossRefGoogle Scholar
  28. 28.
    Gor GY et al (2012) Quenched solid density functional theory method for characterization of mesoporous carbons by nitrogen adsorption. Carbon 50(4):1583–1590CrossRefGoogle Scholar
  29. 29.
    Fairen-Jimenez D, Carrasco-Marin F, Moreno-Castilla C (2007) Adsorption of benzene, toluene, and xylenes on monolithic carbon aerogels from dry air flows. Langmuir 23(20):10095–10101CrossRefGoogle Scholar
  30. 30.
    Kast W, Hohenthanner CR (2000) Mass transfer within the gas-phase of porous media. Int J Heat Mass Transf: 807–823Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Universidad Pontificía BolivarianaMonteríaColombia
  2. 2.Facultad de MinasUniversidad Nacional de ColombiaMedellínColombia
  3. 3.Facultad de IngenieríasUniversidad de La GuajiraRiohachaColombia

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