Vol.:(0123456789) 1 3
Arabian Journal for Science and Engineering
https://doi.org/10.1007/s13369-019-04259-x
RESEARCH ARTICLE - CHEMICAL ENGINEERING
Study of Two‑Phase Nonlinear Advection Dispersion Model
for Displacement Washing of Porous Particles Using OCFE
with Lagrangian Basis
Shelly Arora
1
· Dereje Alemu Alemar
2
· František Potůček
3
Received: 25 April 2017 / Accepted: 15 November 2019
© King Fahd University of Petroleum & Minerals 2019
Abstract
A comprehensive difusion dispersion model has been presented for displacement washing of porous particles. Model equa-
tions have been divided into two phases, namely bulk fuid phase and particle phase. Both the phases have been character-
ized by particle geometry and pore radius of particles. Inter-pore and intra-pore solute concentrations have been related
to Langmuir adsorption isotherm. Nonlinear set of model equations has been solved by using the technique of orthogonal
collocation on fnite elements with Lagrangian basis. Efect of diferent parameters such as Péclet number, bed porosity and
distribution ratio has been shown graphically via breakthrough curves and surface plots. Stability of the numerical tech-
nique has been checked by L
2
and L
∞
norms for diferent values of parameters. Validity of the model on the laboratory-scale
washer has been verifed by comparing experimental and model-predicted values. Applicability of the model has also been
discussed through industrial parameters.
Keywords Orthogonal collocation · Axial dispersion coefcient · Péclet number · Bed porosity · Pore radius of particles ·
Particle geometry
List of Symbols
A Cross-sectional area of drum (m
2
)
A Particle geometry
Bi Biot number
(
k
f
R
KD
F
)
C Concentration of bulk fuid (kg/m
3
)
c
av
Average solute concentration (kg/m
3
)
C
0
Initial solute concentration (kg/m
3
)
C Dimensionless solute concentration (c/C
0
)
Q Inter-particle solute concentration (kg/m
3
)
Q Dimensionless inter-particle solute concentration
(q/C
0
)
n Intra-particle solute concentration (kg/m
3
)
N
0
Initial intra-particle solute concentration (kg/m
3
)
N Dimensionless intra-particle solute concentration
(n/N
0
)
k
f
Film resistance mass transfer coefcient (m/s)
k
1
Forward rate constant (1/s)
k
2
Backward rate constant (1/s)
L Cake thickness (m)
Pe Peclet number (uL/D
L
)
r Variable pore radius of particles (m)
R Pore radius of particles (m)
t Time of washing (s)
u Interstitial velocity (m/s)
z Variable cake thickness (m)
Greek Symbols
β Porosity of particles
ε Porosity of packed bed
γ Variable element size
η Dimensionless pore radius of particles (r/R)
η* Dimensionless variable
θ Dimensionless parameter
(
a(1-)
)
* Shelly Arora
aroshelly@gmail.com
Dereje Alemu Alemar
alemardereje@gmail.com
František Potůček
frantisek.potucek@upce.cz
1
Department of Mathematics, Punjabi University, Patiala,
Punjab 147002, India
2
Department of Mathematics, College of Natural
and Computational Science, Jigjiga University, Jigjiga,
Ethiopia
3
Department of Wood, Pulp and Paper Technology, University
of Pardubice, 53210 Pardubice, Czech Republic