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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
Original
  • 21 Downloads

Abstract

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.

Nomenclature

a1

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

b

Langmuir equilibrium constant

b1

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

c

liquid phase concentration

c0

liquid phase concentration in feed (the infinite bath)

D0

limiting diffusivity at low storage [m2/s]

De

effective concentration-dependent diffusivity[m2/s]

Dp

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) [−]

Fo

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

Fn(λ)

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

K

Henry constant (K = baqs) based on particle volume

km = ma(t)/m(t)

fractional uptake ratio [−]

L

half thickness of the slab

M(λ)

nonlinear parameter defined by eq.(27)

Mc(λ)

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

m

dimensionless exponent in eq. (7b)

ma(t)

mass adsorbed at time t

m(t)

mass adsorbed at equilibrium

n

exponent of the assumed parabolic profile [−]

nopt

optimal exponent of the assumed parabolic profile [−]

n0

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

n(λ)

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

p

chemical potential

q

adsorbed phase concentration

q0

adsorbed phase concentration at equilibrium with c0

qs

saturation limit (Langmuir model)

r

radial coordinate [m]

R

particle radius [m]

t

time [s]

tL

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

t0

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

tsat

saturation time [s]

u

dimensionless adsorbed concentration (u = q/qs)

ua

assumed concentration profile (dimensionless) [−]

us

dimensionless surface concentration defined by eq.(19)

x

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

Greek symbols

β

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

δ

penetration depth [m]

δ0

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

ε

particle porosity [−]

η = x/δ

dimensionless distance [−]

φ(ua(x, t))

residual function of the governing diffusion equation

λ

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)

Subscripts

HBIM

Heat-balance Integral Method

DIM

Double-Integral Method

Abbreviations

DIM

Double-Integral Method

HBIM

Heat-balance Integral Method

Notes

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

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

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