Heat and Mass Transfer

, Volume 55, Issue 2, pp 261–279 | Cite as

A new closed-form approximate solution to diffusion with quadratic Fujita’s non-linearity: the case of diffusion controlled sorption kinetics relevant to rectangular adsorption isotherms

  • Jordan HristovEmail author


A new approximate solution relevant to case of rectangular (Langmuirian) adsorption isotherms has been developed on the basis of the integral-balance method with double integration technique. The solution is based on the model developed by Ruthven for slab and spherical adsorption pellets where the adsorption is controlled by the diffusion in macropores. This model results in a concentration-dependent diffusion coefficient with a quadratic nonlinearity of Fujita’s type. The new approximate solution utilizes the concept of finite penetration depth of the diffusant penetration and almost shockwave shape of the concentration profiles. The solution is based on an assumed parabolic profile with unspecified exponent. The solution analysis allowed determining the optimal exponents of the concentration profile as a function of the nonlinearity parameter of the Fujita’s concentration relationship. The approximate solutions have been compared to existing results from the literature and especially the data published by Ruthven and Crank, for concentration distributions and fractional uptakes in an adsorbent slab pellet.



constant in the non-linear Fujita’s concentration dependence of the diffusivity


Langmuir equilibrium constant


constant in the non-linear Fujita’s concentration dependence of the diffusivity


liquid phase concentration


liquid phase concentration in feed (the infinite bath)


limiting diffusivity at low storage [m2/s]


effective concentration-dependent diffusivity[m2/s]


particle diffusivity[m2/s]

EL(n, λ, t)

mean-squared error of approximation over the penetration depth [−]

eL(n, λ)

time-independent function defined by eq. (55) [−]


Fourier number (Fo = t/t0 = D0t/L2 = [Dpεt/KL2(1 − ε)])[−]


functional relationship defined by eq.(53) [−]


Henry constant (K = baqs) based on particle volume

km = ma(t)/m(t)

fractional uptake ratio [−]


half thickness of the slab


nonlinear parameter defined by eq.(27)


nonlinear parameter used in the solution of Crank (Mc = 1/(1 − λ)2)


dimensionless exponent in eq. (7b)


mass adsorbed at time t


mass adsorbed at equilibrium


exponent of the assumed parabolic profile [−]


optimal exponent of the assumed parabolic profile [−]


optimal exponent defined for the case with λ = 0[−]


exponent of the assumed parabolic profile defined by eq. (53)[−]


chemical potential


adsorbed phase concentration


adsorbed phase concentration at equilibrium with c0


saturation limit (Langmuir model)


radial coordinate [m]


particle radius [m]


time [s]


time required the concentration wave to travel the half of the slab thickness L


characteristic diffusion time (t0 = L2/D0)[s]


saturation time [s]


dimensionless adsorbed concentration (u = q/qs)


assumed concentration profile (dimensionless) [−]


dimensionless surface concentration defined by eq.(19)


distance from the particle (slab) surface) [m]

Greek symbols


small parameter defining the limit of the approximate profile at the front [−]


penetration depth [m]


penetration depth in the linear case for λ = 0[m]


particle porosity [−]

η = x/δ

dimensionless distance [−]

φ(ua(x, t))

residual function of the governing diffusion equation


non-linearity parameter (λ = q0/qs)[−]

\( {\xi}_x=x/\sqrt{D_0t} \)

Boltzmann similarity variable (rectangular coordinates) (slab adsorbent)[−]

\( {\xi}_r=r/\sqrt{D_0t} \)

Boltzmann similarity variable (spherical adsorbent)[−]

\( {\xi}_{xC}=x/2\sqrt{\left({D}_p/K\right)t} \)

similarity variable expressed through the Henry constant [−]

Θ = (1 − λu)

dimensionless variable (eq.7b)



Heat-balance Integral Method


Double-Integral Method



Double-Integral Method


Heat-balance Integral Method



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Authors and Affiliations

  1. 1.Department of Chemical EngineeringUniversity of Chemical Technology and MetallurgySofiaBulgaria

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