Volume averageing

Journal Pre-proof
Volume averaging theory (VAT) based modelling for longitudinal mass
dispersion in structured porous medium with porous particles
Chen Yang Rong Huang Yixiong Lin Ting Qiu
PII:
S0263-8762(19)30518-0
DOI:
https://doi.org/doi:10.1016/j.cherd.2019.10.048
Reference:
CHERD 3883
To appear in:
Chemical Engineering Research and Design
Received Date:
1 September 2018
Revised Date:
17 September 2019
Accepted Date:
31 October 2019
Please cite this article as: Yang, C., Huang, R., Lin, Y., Qiu, T.,Volume averaging theory (VAT)
based modelling for longitudinal mass dispersion in structured porous medium with porous
particles, Chemical Engineering Research and Design (2019),
doi: https://doi.org/10.1016/j.cherd.2019.10.048
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© 2019 Published by Elsevier.
Research Highlights
Highlights

Mass transfer in structured porous media with porous particles was upscaled.

A macroscopic solute transport equation was proposed by using VAT.

A new correlation of longitudinal mass dispersion was presented.

A comparison between random and structured porous media correlations was
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*Manuscript
Click here to view linked References
1
Volume averaging theory (VAT) based modelling for longitudinal
2
mass dispersion in structured porous medium with porous particles
3
Chen Yang, Rong Huang, Yixiong Lin, Ting Qiu*
4
Correspondence author: [email protected] (T. Qiu)
5
Fujian Universities Engineering Research Center of Reactive Distillation Technology,
6
College of Chemical Engineering, Fuzhou University, Fuzhou 350116, Fujian, China
Abstract
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of
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In this study, to investigate the mass transport characteristics in a structured
9
porous medium with porous particles under an interfacial concentration discontinuity,
10
a macroscopic solute transport equation is proposed based on the volume averaging
11
theory. A typical three-dimensional geometry (a body center cubic arrangement of
12
spheres) was chosen as the representative elementary volume. The corresponding
13
closure problem was solved to obtain the longitudinal mass dispersion. Based on the
14
numerical results, a new correlation of the longitudinal mass dispersion for a
15
structured porous medium with porous particles is presented. Through a comparison
16
with the correlation of the longitudinal mass dispersion for a random porous medium,
17
it can be determined that, for a mechanical dispersion, the dependence of a random
18
porous medium and that of a structured porous medium on the Péclet number Pe are
19
linear and quadratic, respectively. Furthermore, it was also determined that for a
20
holdup dispersion, a structured porous medium is less dependent on the ratio of
21
diffusivity of the fluid phase to the diffusivity of the porous particle phase. In addition,
22
it is more dependent on the inverse ratio of the solubilities of the solute in the fluid
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1
and in the catalyst particles than in a random porous medium.
2
Keywords: Volume averaging theory (VAT); Longitudinal mass dispersion;
3
Structured porous medium; Porous particles; Péclet number
4
1. Introduction
Owing to its convenience and efficiency, a porous medium model is important
6
for understanding the mass transport characteristics in many technological and
7
environmental processes (Bear, 1988; Huang et al., 2010; Sano and Nakayama, 2012;
8
Yang et al., 2015). To implement a porous medium model for practical applications,
9
an effective diffusion coefficient that is affected by the geometrical and hydraulic
10
factors should be obtained in advance. With respect to the hydraulic flow of a liquid in
11
a porous medium, the dispersion resulting from the hydrodynamic mixing of the fluid
12
particles passing through pores plays an important role in dealing with the transport
13
processes.
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Based on an asymptotic analysis in a bed of fixed spheres, Koch and Brady
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(1985) proposed several physical mechanisms causing dispersion, namely, mechanical
16
dispersion, boundary-layer dispersion, and holdup dispersion. The latter two
17
mechanisms are also a type of non-mechanical dispersion. Chai et al. (2016)
18
summarized
19
experimental, numerical, and analytical (or semi-analytical).
three
categories
related
to
dispersion
measurement
methods:
20
By introducing a step change in the concentration of a dilute solute, Fried and
21
Combarnous (1971) obtained numerous experiment results on the mass dispersion in
22
packed beds of nonporous particles. This was achieved by fitting the exponential
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Page 3 of 37
profiles based on Fick’s law at the macroscopic scale with the resultant concentration
2
profile downstream from the inlet. For a fixed bed of random particles, Kandhai et al.
3
(2002) used pulsed field gradient nuclear magnetic resonance (PFG-NMR) to obtain
4
the longitudinal mass dispersion of both porous and nonporous particles. Moreover,
5
their experiment results also indicated the importance of a holdup dispersion. For
6
another experimental method associated with the mass dispersion of gases and liquids
7
in a packed bed, see the excellent study by Delgado (2006). Nevertheless, the problem
8
related to the experimental method is that the measured results are insufficient to
9
provide a detailed interpretation of the dispersion mechanisms.
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Using a numerical method, Blunt et al. (2013) computed the mass dispersion in
11
different rock samples, and obtained the physical geometries through pore-scale
12
imaging. Yang et al. (2018) recently utilized a D2Q9 model with a
13
multi-relaxation-time (MRT) collision operator, which is a type of lattice Boltzmann
14
method used to calculate the longitudinal mass dispersion of a porous medium with
15
randomly positioned particles. A random placement (RP) method was used to
16
reconstruct the physical geometry of the porous medium with random and
17
nonoverlapping particles. Moreover, a general correlation of the longitudinal mass
18
dispersion validated through the experiment data under a wide-ranging porosity was
19
proposed. Although the numerical method has been considered the most robust, it is
20
normally time-consuming owing to the difficulty in obtaining the microscopic
21
geometry through laboratory measurements, as pointed out by Raeini et al. (2014).
22
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To elucidate the mass transport mechanisms in a porous medium, analytical and
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Page 4 of 37
semi-analytical methods have been extensively applied. Quintard and Whitaker (1994)
2
developed a macroscopic model for a homogeneous porous medium using a volume
3
averaging method. They obtained analytical solutions to the longitudinal mass
4
dispersion and transverse mass dispersion for a few one-dimensional and
5
two-dimensional unit cells. Furthermore, Neculae et al. (2002) used the volume
6
averaging method to obtain the solute dispersion in a columnar dendritic mushy zone,
7
thereby improving the mass transport description during solidification modelling. To
8
investigate the effects of both heterogeneous and homogenous chemical reactions on
9
the mass transport in a porous medium, Valdés-Parada et al. (2011) applied upscaling
10
processes to achieve a mass dispersion under consideration of heterogeneous and
11
homogeneous chemical reactions, and affirmed that the increase in the Thiele moduli
12
results in a decrease in the mass dispersion. A summary of the related studies can be
13
seen in Table 1.
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Table 1 Summary of the mass transfer process in a porous medium based on VAT
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Reference
Geometry
Quintard and Whitaker
Interfacial condition
Constant concentration or zero
Stratified system
(1994)
mass flux
Neculae et al. (2002)
Dendritic structure
Zero mass flux
Valdés-Parada et al.
In-line arrangement of
Homogeneous and
(2011)
squares or spheres
heterogeneous reactions
Heterogeneous reaction coupled
Yang et al. (2015)
Stratified system
with temperature
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In-line arrangement of
Reversible linear heterogeneous
spheres
chemical reaction
Qiu et al. (2017)
It should be noted that in the upscaling studies on mass dispersion in a porous
2
medium described above, all solid matrixes were assumed to be impermeable.
3
However, for practical applications (Gerke and Genuchten, 1993; Nakayama and
4
Sano, 2013), a porous medium with permeable particles (such as porous pellets,
5
which have an interfacial concentration discontinuity) is more frequently encountered.
6
Therefore, we prefer the use of the volume averaging theory (VAT) to obtain a
7
macroscopic solute transport equation for a porous medium with permeable particles.
8
Moreover, a three-dimensional structured geometry is chosen as representative
9
elementary volume (REV). By solving the corresponding closure problems, the
10
longitudinal mass dispersion is calculated to elucidate the mass transport mechanisms
11
in a structured porous medium with permeable particles.
12
2. Volume averaging theory
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Fig. 1 shows a schematic diagram of the mass transfer in a porous medium
14
composed of a void and a solid matrix. Fluid passes through the permeable void of the
15
solid matrix. Inside the permeable particles, no fluid flow is considered. Therefore,
16
there are two phases (a fluid phase in the void and a porous particle phase) when
17
considering the solute diffusion in the permeable particles. Moreover, owing to the
18
difference in solubility between the fluid phase and the porous particle phase, a solute
19
concentration discontinuity occurs at the interface (Koch and Brady, 1985). For the
20
interphase mass transfer of single drops in an immiscible liquid, Yang and Mao (2005)
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Page 6 of 37
also considered the effect of the interfacial mass discontinuity on the overall mass
2
transfer coefficient.
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Fig. 1. Schematic diagram of mass transfer in porous media
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To investigate the convective mass transport characteristics of a porous medium
6
with permeable catalyst particles, the governing equations for the mass transfer
7
system without the chemical reactions of the mass, momentum, and solute
8
concentration are expressed as follows:
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
   (  u)  0
t
 (  u)
   (  uu)  p    σ
t
C f
t
   (C f u)    ( D f C f )
Cm
   ( DmCm )
t
(1)
in V
in Vf
(2)
in Vf
(3)
in Vm
(4)
9
where  is the local mass density; u is the fluid velocity vector; p is the
10
pressure; σ is the viscous stress tensor; C is the solute concentration; D is the
11
related diffusion coefficient; and subscripts f and m refer to the fluid and porous
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1
particle phases, respectively. Note that Dm is the intraparticle diffusivity of a solute
2
inside porous particles and is normally affected by the pore structure of the particles,
3
such as the porosity and tortuosity (Zalc et al., 2004; Xuan et al., 2010; Lin et al.,
4
2019; Zheng et al., 2019). In addition, V is the total volume, whereas Vf is the fluid
5
volume around porous particles.
At the interface between the fluid phase and porous particle phase, Koch and
7
Brady (1985) used the continuity of the flux and solubility conditions, which is
8
indicated as follows:
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n fm  D f C f  n fm  DmCm
C f  mCm
at Afm
at Afm
(5a)
(5b)
where n fm is the normal unit vector from the fluid phase to the porous particle phase;
10
m is the ratio of the solubilities of the solute in the fluid and in the catalyst particles;
11
and Afm is interfacial surface area between the fluid and solid phases.
12
2.1. Transformation of governing equations
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9
Chastanet and Wood (2008) investigated the mass transfer process in a
14
two-region medium, although a continuous mass concentration was assumed, namely,
15
m = 1. Because m is generally different from unity, the discontinuous mass
16
concentration in the interface, as shown in Eq. (5), brings about difficulty in the
17
upscaling procedures, and thus macroscopic equations are obtained from microscopic
18
equations based on the VAT. Based on our previous study (Yang et al., 2018), the
19
proposed correlation shows better agreement with the experiment data for the case of
20
m = 5. Zhou et al. (2015) resolved the concentration under a discontinuous condition
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during the gas–solid adsorption process through the iteration of the boundary
2
conditions. In this way, the concentrations in the fluid and porous particle phases were
3
determined at the same time by solving two mass transfer equations directly. However,
4
it should be mentioned that this method requires more computing resources and is
5
inapplicable during upscaling procedures. Therefore, the governing equations, namely,
6
Eqs. (1)–(4), are transformed to simplify the upscaling processes. In this study, the
7
concentration transformation method first proposed by Yang and Mao (2005) was
8
used to make the concentration field continuous across the interface. For this purpose,
9
the concentration transformations are as follows:
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C f  C f
m,
(6)
11
Cm  Cm m ,
(7)
14
15

C f
 t
m

lP
13
Thus, Eqs. (3) and (4) can be rewritten separately as follows:
   (C f
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10
Cm

 t m

mu)    ( mD f C f ) ,
   ( Dm
m Cm ) .
(9)
Correspondingly, the transformed boundary conditions are as indicated below:
n fm  mD f C f  n fm  Dm
C f  Cm
16
(8)
m Cm
(10)
(11)
In Eqs. (10) and (11), it can be clearly seen that, after the implementation of the
17
concentration transformation method, both the interfacial concentration and the mass
18
flux become continuous across the interface. Although there are some other
19
transformation forms, Yang and Mao (2005) pointed out that such form has no effect
20
on the numerical solutions.
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Page 9 of 37
2
For further manipulations, the microscopic governing equations for a solute
transport in a porous medium with porous particles are obtained as follows:
C f
   (C f u)    ( Df C f )
t f
3
Cm
   ( Dm Cm )
tm
4
5
(12)
in Vm
(13)
where
6
t f  t
7
Df  D f m
m
tm  t m
and
Dm  Dm
and
u  u m
m
(14a, 14b)
(14c, 14d)
(14e)
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9
in Vf
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2.2. Upscaling procedures
To obtain the longitudinal mass dispersion, the volume averaging method (Cheng,
11
1979; Nakayama 1995; Whitaker, 1999) was used. Macroscale concentrations are
12
defined over the representative elementary volume, shown in Fig. 2, as follows:
14
15
16
1
V

lP
f 
Vf
f
 f dV ,
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f

1
Vf
 f f  f

Vf
 f dV ,
(15a, 15b)
f
,
(15c)
where  f = V f V .
In addition, the decomposition in terms of the average intrinsic concentration and
17
the spatial deviation concentration based on Gray’s (1975) spatial decomposition are
18
introduced as follows:
C f  C f
f
19
Cm  Cm
m
20
21
 C f
(16a)
 Cm
(16b)
In the same way, the decomposition of the velocity can be expressed in terms of
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1
the intrinsic average velocity and the spatial deviation velocity, which is defined as
2
u  u
 u
(17)
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3
5
Fig. 2. Averaging volume for fluid and porous particle phases in the REV
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Based on the assumption of a scale separation, usually expressed as
L , the macroscopic solute transport equation of the fluid phase and
6
l f , lm
r0
7
porous particle phase are given as follows:

  f C f
f
t f
8
  

f
u
f
C f
f
     D (  C
f
f
   uC f 

  m Cm
9
tm
m
     D (  C

1

V

Amf
m
m
m
m

1
V

f
1
V
Amf

A fm
f

1
V

A fm

n fmC f dA) 
 , (18)
n fm  Df C f dA

n mf Cm dA) 
.
(19)
n mf  Dm Cm dA
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1
2
Moreover, the boundary conditions are given as

n fm  Df  C f
C f
3

f

 C f  n fm  Dm  Cm
f
 C f  Cm
m

 Cm ,
 Cm .
m
(20)
(21)
To close the macroscopic equations, we can obtain the closure problem to
5
eliminate the fluctuating variables. By applying decomposition Eqs. (16a) and (16b)
6
in the microscale governing Eqs. (12) and (13) and subtracting the macroscale Eqs.
7
(18) and (19), the corresponding closure problems can be obtained. Normally, the
8
governing equations of the fluctuating quantities can be further simplified through the
9
following inequalities:
u C f
11
13

-p
 Df
V
 f 1  
  Dm Cm


n fmC f dA  ,
Afm


 f 1   uf Cf  ,
 Dm
 V
 m1  

Amf

n mf Cm dA  .

(22)
(23)
(24)
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12

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
  Df C f
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For a diffusive heat transport through a two-phase material, Monye (1997) used
14
the volume averaging approach to obtain a local thermal non-equilibrium model, and
15
to compare the results of an unsteady closure and a quasi-steady closure. The
16
unsteady closure results in an extra expression of the phase exchange term and
17
conductive terms as temporal convolutions, although (surprisingly) the quasi-steady
18
closure gives satisfactory results. Therefore, the closure problems are usually assumed
19
quasi-steady, as described in related studies by Moyne (1997), Wood et al. (2003), and
20
Davit et al. (2012). Moreover, the simplified steady governing equations of the spatial
21
fluctuations are indicated as follows:
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1
u C f +u  C f



   Df C f 

0    Dm Cm 
2
3
f
 m1
V

Amf
 f 1
V

A fm
n fm  Df C f dA ,
n mf  Dm Cm dA .
(25)
(26)
2.3. One-equation model
In practical applications, it is more convenient to represent the concentrations in
5
the fluid and porous particle phases in terms of a single concentration and a Darcy
6
deviation. For heat transfer in a porous medium, a similar approach was given by
7
Quintard and Whitaker (2000). Based on Eqs. (18) and (19), a one-equation model is
8
constructed. These new compositions are given through the following:
Cm
m
10
where
C
14
15
(28)
1
V

V
C dV   f C f
f
  m Cm
m
.
(29)
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13
 C   Cˆ m ,
represents the spatial average concentration defined explicitly by
C 
12
(27)
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11
 C   Cˆ f ,
re
C f
f
9
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By substituting Eqs. (27) and (28) into Eqs. (18) and (19), the following is
obtained:
 m   C

f
 m f 
 t   f u  C  
m

D


D
    f Df   m Dm   C  + f  n fmC f dA  m  n mf Cm dA
V Afm
V Amf




Cˆ f  m Cˆ m
f
   uC f   m f

  f u Cˆ f     f Df Cˆ f   m Dm Cˆ m 
t
m t




16
(30)
17
For the local mass equilibrium conditions, the last four terms on the right side of
18
Eq. (30) are sufficiently small to be discarded, as pointed out by Quintard and
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Page 13 of 37
1
Whitaker (2000). Furthermore, the spatial deviation concentrations in terms of the
2
closure variables are as follows:
C f  b f  C  ,
3
Cm  bm  C  .
4
Therefore, the conductive terms in Eq. (30) can be expressed as
14
15
18
n fmC f dA 

(33)

-p
(34)
re
By substituting Eqs. (33) and (34) into Eq. (30), the one-equation model can be
given as
 m   C

f
m


f

 t   f u  C      f D  C   ,
m

(35)
where
 f D   f Dstag + f Ddis =  f Df   m Dm  I 
 D  D 
f
m
V

A fm
n fmb f dA  ub f . (36)
According to the transformation rules given by Eqs. (14a)–(14e), the general
form of the one-equation model can be given as
 C
  *f u
t
16
17
A fm
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13

 uCf   ub f  C =    f Ddis  C  .
11
12
Df   m Dm   C  +
In addition, the dispersive transport term can be determined as
8
10
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9
Df
Dm
n mf Cm dA=
V
V Amf
.


Df  Dm 

  f Df   m Dm  I 
Afm n fmb f dA  C    f Dstag  C 
V



7
(32)
lP
5
(31)
f
 C      *f D  C  ,
(37)
where
  f


 D f  m Dm 
f

m

 f D   f Dstag + f Ddis =  f D f   m Dm I  
n fmb f dA   f ub f

A
fm
V


f
, (38)
13 / 36
Page 14 of 37
m f
 f 
1
m f 
m
m
 m 
2
m
m
m
m
m f 
,
(39)
.
(40)
Note that when m is infinitive, dispersion tensor Ddis in Eq. (38) can be used in an
4
impermeable case.
5
2.4 Closure problem
ro
of
3
As shown in Eq. (38), the theoretical prediction of the mass dispersion tensor
7
requires the closure variable b f . Based on the closure variables defined in Eqs. (31)
8
and (32), the closure problem is given as
11
12
13
14
15
16
17
re
lP
10
u b f +u     Df b f  
0     Dm b m  
 m1
V
Jo
ur
na
9
-p
6

Amf
 f 1
V

A fm
n fm  Df b f dA ,
n mf  Dm b m dA .
(41)
(42)
The boundary conditions are as follows:
b f =b m
n fm  Df b f =n fm  Dm b m  n fm  Df  Dm 
at Afm,
(43)
at Afm,
(44)
i  1, 2,3 .
(45)
The periodicity is as follows:
b f r 
i
  b f r  ,
bm  r 
i
  bm  r  ,
Finally, the average is
bf
f
 0,
bm
m
0
(46)
18
As shown in Eqs. (41) and (42), these are integro-differential closure equations.
19
To handle problems involving integrals of the closure variables as the source terms on
14 / 36
Page 15 of 37
the right-hand side of Eqs. (41) and (42), the decomposition method proposed by
2
Quintard and Whitaker (1994) and Quintard et al. (1997) was used. However, it
3
should be mentioned that the present approach is capable of solving the difficulty in
4
the upscaling procedures resulting from a discontinuous mass concentration at the
5
interface, as shown in Eq. (5b). Under the constraints of Eq. (46), Eqs. (41) and (42)
6
can be easily solved under the boundary conditions of Eqs. (43) and (44) in a periodic
7
unit cell, where the spatially periodic boundary conditions at the inlet and outlet of the
8
fluid, as shown in Eq. (45), were applied. Subsequently, based on the given results of
9
the closure variable b f , the dispersion tensor can be determined through Eq. (38).
-p
3. Validation and physical geometry
re
10
ro
of
1
As shown in Eq. (38), the dispersion tensor includes nine elements in three
12
directions. To validate the VAT approach presented in the present study, the
13
longitudinal mass dispersion Ddisxx , which can be obtained using the spatial deviation
14
of velocity u and the value of closure variable b f , both in the x direction, is
15
compared with the results of several previous studies under the variation of the Péclet
16
number. Note that the REV is the two-dimensional in-line array of cylinders shown in
17
Fig. 3 and that the porosity is 0.48. Moreover, the Péclet number is calculated based
18
on the particle diameter rather than the equivalent particle diameter, which is defined
19
by 6 aV . The equations related to the closure problem are solved using the COMSOL
20
Multiphysics package. As the convergence criteria, the residuals of all equations are
21
less than 10-5. Furthermore, it should be mentioned that all numerical results are
22
independent of the grid size. As demonstrated in Fig. 3, the comparison between the
Jo
ur
na
lP
11
15 / 36
Page 16 of 37
present study and that by Eidsath et al. (1983) was conducted for the same REV,
2
allowing the implementations of all parameters in the VAT approach to be confirmed.
3
The agreement between these two studies was found to be satisfactory. However, it
4
should be noted that the VAT approach proposed by Eidsath et al. is only applicable
5
for a porous medium composed of impermeable particles. Because the only
6
experiment measurements of the longitudinal mass dispersion for a spatially periodic
7
porous medium applied are those of Gunn and Pryce (1969), the present VAT
8
approach was validated by comparing the numerical results with limited experiment
9
data. The figure clearly shows that our numerical results are in accordance with the
-p
experiment data within a large range of Péclet number.
Jo
ur
na
lP
re
10
ro
of
1
11
12
Fig. 3. Comparison of the present study with the numerical results by Eidsath et al.
13
and experiment data by Gunn and Pryce
16 / 36
Page 17 of 37
Based on the VAT approach, a representative elementary volume is necessary to
2
obtain the effective coefficients. In the present study, we attempted to obtain the
3
longitudinal mass dispersion in a structured porous medium with porous particles. In
4
the literature, there are several classic three-dimensional sphere-packed structures,
5
including those reported by Yang et al. (2010), Bu et al. (2014), and Wang et al.
6
(2017), such as simple cubic, body center cubic (BCC), and face center cubic ordered
7
architectures. Owing to its applicability to a large porosity variation and simplicity, a
8
BCC arrangement of spheres is chosen as the representative elementary volume, as
9
illustrated in Fig. 4.
10
11
12
13
Jo
ur
na
lP
re
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of
1
Fig. 4. REV in BCC arrangement of spheres
4. Results and discussion
As indicated in Eq. (38), the expression of mass dispersion is given as
14
Ddis = ub f
f
.
(47)
15
Moreover, it can be seen from Eq. (37) that the intrinsic averaged velocity in the
16

present study is  f u . Therefore, the Péclet number in the present study is defined
17
as follows:
f
17 / 36
Page 18 of 37
Pe 
1
2
 f u d p
Df
,
(48)
where d p is the particle diameter.
In this section, we describe the correlation of longitudinal mass dispersion in a
4
structured porous medium with porous particles. Note that the longitudinal mass
5
dispersion is commonly expressed as DL in the literature, which is equal to Ddisxx in
6
the present study. Based on an analytical analysis conducted by Koch and Brady
7
(1985), the full form of the longitudinal mass dispersion can be written as follows:
ro
of
3
n
(49)
-p
2
n
 D f   1  3 n4
DL
n1
  m Pe  b Pe ln  Pe    h 
   Pe
Df
 Dm   m 
Note that there are three terms in Eq. (49), namely, the contributions of mechanical
9
dispersion, boundary-layer dispersion, and holdup dispersion. It should also be noted
10
that  m , b , and  h are the coefficients of the three counterparts of longitudinal
11
mass dispersion, which are the functions of porosity. In Eq. (49), it can easily be seen
12
that the longitudinal mass dispersion varies with several parameters, namely, ε, Pe,
lP
Df
Jo
ur
na
13
re
8
Dm
, and m. In the following subsections, numerical simulations related to the effects
Df
14
of
15
and ε to obtain a general correlation of longitudinal mass dispersion are described.
16
4.1 Effects of m variation
Dm
and m on the longitudinal mass dispersion performed under variations of Pe
17
In the present study, the aim is to obtain the longitudinal mass dispersion in a
18
structured porous medium with porous particles. Because a concentration
19
discontinuity occurs at the interface between the fluid phase and the porous particle
18 / 36
Page 19 of 37
phase, the effect of an interfacial concentration discontinuity on the longitudinal mass
2
dispersion is investigated. Moreover, the interfacial mass discontinuity can be
3
represented using the solubility ratio m. Here, m 1  0 and m 1  0 represent two
4
distinct situations, namely, fixed beds with impermeable and permeable particles. In
5
Fig. 5, DL D f is plotted against the variation of Pe for ε = 0.35 and D f Dm  10 .
6
Three solubilities, namely, m = 1, 2, and 5, are considered. Furthermore, similar
7
numerical simulations were conducted for ε = 0.45 and 0.55, as indicated in Figs. 6
8
and 7. In these three figures, it can be observed that the longitudinal mass dispersion
9
of a porous medium with permeable particles is larger than that of a porous medium
10
with impermeable particles. In addition, the difference indicates the significance of a
11
holdup dispersion. This is consistent with the experiment results of Kandhai et al.
12
(2002), who used PFG-NMR to measure the longitudinal mass dispersion of a porous
13
medium with porous and nonporous particles. It can clearly be seen in these three
14
figures that with a decrease in m, the holdup dispersion contribution becomes more
15
significant, which is in accordance with the expression of the holdup dispersion.
16
Moreover, it can also be found that, for the structured arrangement, the longitudinal
17
mass dispersion of a porous medium with permeable particles is close to that of a
18
porous medium with impermeable particles when m = 5. For a random arrangement of
19
permeable particles, however, Yang et al. (2018) pointed out that the longitudinal
20
mass dispersion of a porous medium with permeable particles for m = 10 approaches
21
that of a porous medium with impermeable particles.
Jo
ur
na
lP
re
-p
ro
of
1
19 / 36
Page 20 of 37
ro
of
-p
Fig. 5. Effect of m variation on longitudinal mass dispersion for ε = 0.35 and
Df
Dm
=10
Jo
ur
na
lP
2
re
1
3
4
Fig. 6. Effect of m variation on longitudinal mass dispersion for ε = 0.45 and
Df
Dm
=10
20 / 36
Page 21 of 37
ro
of
-p
1
Fig. 7. Effect of m variation on longitudinal mass dispersion for ε = 0.55 and
3
4.2 Effect of D f Dm variation
Df
Dm
=10
lP
re
2
As shown in the expression of a holdup dispersion, D f Dm is also an important
5
parameter. Hence, DL D f was plotted against the variation of Pe for ε = 0.35 and m
6
= 1 in Fig. 8. The intraparticle diffusion coefficient of a porous particle is mainly
7
dependent on the microstructure of a porous particle. Hussain et al. (2015) proposed a
8
new algorithm, RGMMP, based on a connection of the macro-meso pores to generate
9
the microstructure of porous particles with hierarchical pores, such as the building
10
materials, ion exchanger resin, and molecular sieve. Based on the macro-meso
11
structures, the corresponding numerical results indicate that the connection between
12
the macro and meso pores is crucial to predict the intraparticle diffusion coefficient.
13
Moreover, Yoshida and Taktsuji (2000) experimentally obtained the intraparticle
Jo
ur
na
4
21 / 36
Page 22 of 37
diffusion coefficients of acetoc and lactic acids in a weakly basic ion exchanger, and
2
pointed out that Df is normally one- or two-orders of magnitude higher than Dm.
3
Therefore, three ratios of the molecular diffusion coefficient of a solute to the
4
intraparticle diffusion coefficient, namely, D f Dm  2 , 5 and 50, are considered.
5
Furthermore, similar numerical simulations were conducted for ε = 0.45 and 0.55, in
6
Figs. 9 and 10. As shown in these three figures, it can be seen that the increase in
7
D f Dm results in an increase in the longitudinal mass dispersion. The reason for this
8
is consistent with the interpretation given by Maier et al. (2000), who claimed that a
9
holdup dispersion stems from dead-end pores, recirculation cells, and other regions of
10
zero fluid velocity. However, it should be noted that compared with solubility m, the
11
longitudinal mass dispersion is less sensitive to D f Dm .
Jo
ur
na
lP
re
-p
ro
of
1
12
13
Fig. 8. Effect of D f Dm variation on longitudinal mass dispersion for ε = 0.35 and m
14
=1
22 / 36
Page 23 of 37
ro
of
-p
1
Fig. 9. Effect of D f Dm variation on longitudinal mass dispersion for ε = 0.45 and m
3
=1
Jo
ur
na
lP
re
2
4
5
Fig. 10. Effect of D f Dm variation on longitudinal mass dispersion for ε = 0.55 and m
6
=1
23 / 36
Page 24 of 37
1
4.3 Correlation of longitudinal mass dispersion
To investigate the macroscopic mass transport characteristics of a solute in a
3
porous medium, the prediction of the longitudinal mass dispersion coefficient is
4
important. There are no existing correlations of a structured porous medium that are
5
capable of predicting the longitudinal mass dispersion in a structured porous medium
6
with porous particles with an interfacial concentration discontinuity. Therefore, based
7
on the numerical results described in the previous two subsections, we aim to obtain a
8
general correlation. According to Eq. (49), the unknown values of the parameters
9
(  m , b ,  h ), n1, n2, n, and n4 were obtained using the least-squares method and
-p
are given as follows:
m  (-0.0421  3.449 10-4 )  ln    (0.0134  2.883 10-4 )
(50)
b  (0.0355  2.80 104 ) /  2  (0.00927  0.00163)
(51)
h  (0.261  3.274  1017 ) /  2  (0.684  1.908  1016 )
(52)
n1  2.0
(53)
n2  0.122
(54)
n3  1.723
(55)
n4  1.521
(56)
11
Jo
ur
na
lP
re
10
ro
of
2
Furthermore, the consistency of the proposed correlation with the calculated
12
results was checked for ε = 0.45. In Fig. 11, comparisons of the calculated results and
13
the fitting results are conducted. As this figure shows, the coefficients of
14
determination R2 in these six cases are larger than 0.98, indicating that the predicted
15
results stemming from the general correlation are in good agreement with the
24 / 36
Page 25 of 37
calculated results.
re
-p
ro
of
1
3
lP
2
Fig. 11. Comparisons of calculated results and fitting results for ε = 0.45
For a random porous medium with permeable particles, Yang et al. (2018) used a
5
lattice Boltzmann simulation to calculate the asymptotic longitudinal mass dispersion
6
and presented the following correlations:
7
Jo
ur
na
4
D
DL
 m Pe  b Pe ln  Pe    h  f
D
Df
 p



0.241
1
 
m
0.95
Pe2
(57)
m  (1.110  0.0268)   2  (0.00976  0.00101)
(58)
b  (3.099  0.312)  ln( )  (3.287  0.168)  1   
(59)
h  (0.00868  3.075 106 ) /  2  (0.00852  9.374 10 6 )
(60)
8
By comparing Eq. (49) with Eq. (57), it should be noted that, for a mechanical
9
dispersion contribution, the dependence of a random porous medium and a structured
25 / 36
Page 26 of 37


1
2
porous medium on the Péclet number are   Pe  and  Pe , respectively. Based
2
on the VAT, Quintard and Whitaker (1994) obtained the analytical solutions of
3
mechanical mass dispersion for the three cases of a structured porous medium, and
4
2
found that their dependence on the Péclet number is  Pe . Therefore, it can be
5
concluded that for mechanical dispersion, a structured porous medium exhibits a
6
higher dependence on the Péclet number than that of a random porous medium.
7
Conversely, the holdup dispersion of a structured porous medium is less dependent on
8
the Péclet number than that of a random porous medium. Furthermore, a holdup
9
dispersion is also influenced by D f Dm and 1 m . However, it is interesting to note
10
that, for a holdup dispersion, a structured porous medium is less dependent on
11
D f Dm and more dependent on 1 m than a random porous medium.

re
-p
ro
of

In conclusion, a general correlation of the longitudinal mass dispersion of a
13
structured porous medium with porous particles was established, which can be
14
convenient and useful for macroscopic numerical simulations of the mass transfer
15
process. Furthermore, it should also be noted that the macroscopic governing
16
equations of a mass transfer in a porous medium with porous particles can be applied
17
to the analysis and optimization of catalytic packings (Ding et al., 2015) and packed
18
bed reactors (Tanwar et al., 2018), among other areas.
19
5. Conclusions
Jo
ur
na
lP
12
20
Based on the VAT, a macroscopic solute transport equation for a porous medium
21
with porous particles and an interfacial concentration discontinuity was established.
26 / 36
Page 27 of 37
1
For the difficulty resulting from the interfacial concentration discontinuity during the
2
upscaling process, a concentration transformation method was utilized. To obtain a
3
longitudinal mass dispersion, a BCC arrangement of the spheres was chosen as the
4
REV. By solving the closure problem, the effects of D f Dm and m on the
5
longitudinal mass dispersion were investigated under variations of ε and Pe. Based on
6
the results achieved in this study, the following conclusions can be drawn:
(1) The concentration transformation method used herein is effective at handling
8
the interfacial concentration discontinuity during the upscaling process. Furthermore,
9
a resolution of the closure problem was proposed to obtain the longitudinal mass
-p
10
ro
of
7
dispersion.
(2) The variations of D f Dm and m exhibit an adverse influence on the holdup
12
dispersion. Moreover, the holdup dispersion is more sensitive to m than D f Dm . It
13
can also be seen that the decrease in m significantly augments the longitudinal mass
14
dispersion, indicating the significance of a holdup dispersion.
lP
Jo
ur
na
15
re
11
(3) A general correlation of the longitudinal mass dispersion for a structured
16
porous medium with an interfacial concentration discontinuity was obtained based on
17
the least-squares method. In comparison with the existing correlation of a random
18
porous medium, it is interesting to note that, for a mechanical dispersion, the
19
dependence of a random porous medium and a structured porous medium on the
20
2
Péclet number are   Pe  and  Pe , respectively. Furthermore, was also found
21
that for a holdup dispersion, a structured porous medium is less dependent on
22
D f Dm and more dependent on 1 m than a random porous medium.


27 / 36
Page 28 of 37
Nevertheless, the current approach is only applicable for a single-phase fluid
2
system with a single component. Because two-phase flows with multiple components
3
in a porous medium appear in numerous chemical engineering systems, such as
4
chemical reactors and the transport used in petroleum reservoirs, we will address this
5
problem in a forthcoming study using the ideas put forth by Soulaine et al. (2011),
6
who dealt with the upscaling of a multi-component two-phase flow in a porous
7
medium.
8
Acknowledgements
-p
ro
of
1
The authors would like to express their sincere thanks to the National Natural
10
Science Foundation of China (Project Nos. 91534106 and 21506032) for financially
11
supporting this study. The authors would also like to thank Prof. Gérald Debenest of
12
Institut de Mécanique des Fluides de Toulouse (IMFT), France and Dr. Jianwei Guo
13
of the School of Mechanics and Engineering, Southwest Jiaotong University for their
14
valuable suggestions.
15
Nomenclature
Jo
ur
na
lP
re
9
aV
interfacial surface area per unit cell volume (m-1)
Afm
interfacial surface area between the fluid and solid phases (m2)
b
closure variable (-)
C
solute concentration (mol/m3)
D
mass diffusivity (m2/s)
28 / 36
Page 29 of 37
D
effective mass diffusivity tensor (m2/s)
dp
particle diameter (m)
I
unit matrix (-)
L
macroscopic characteristic length (m)
i
i = 1, 2, 3, lattice vectors (m)
microscopic characteristic length (m)
m
ratio of the solubilities of the solute in the fluid and in the catalyst
ro
of
l
particles (-)
normal unit vector from the fluid to the catalyst particle surface (-)
-p
n fm
coefficients in Eq. (47) (-)
p
pressure (Pa)
Pe
Péclet number (-)
r
position vector (m)
ro
characteristic length related to averaging volume
R2
coefficient of determination
Jo
ur
na
lP
re
n
t
time (s)
u
fluid velocity vector (m/s)
V
volume (m3)
x, y, z
Cartesian coordinates (m)
Greek letters
ε
porosity (-)

density (kg/m3)
29 / 36
Page 30 of 37
σ
viscous stress tensor (Pa)

arbitrary variable

coefficients in Eq. (34)
Special symbols
bulk average
f ,m
intrinsic average
deviation from intrinsic average

transformation form defined in Eqs. (14)
ˆ
large-scale spatial deviation defined in Eqs. (27) and (28)
-p
Subscripts and superscripts
boundary-layer
re
b
ro
of

dispersion
f
fluid phase
h
holdup
L
longitudinal mass dispersion
m
mechanical or porous particle phase
p
pore
Jo
ur
na
lP
dis
stag
stagnant
1
2
3
30 / 36
Page 31 of 37
1
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Bu, S.S., Yang, J., Zhou, M., Li, S.Y., Wang, Q.W., Guo, Z.X., 2014. On contact point
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Davit, Y., Wood, B. D., Debenest, G., Quintard, M., 2012. Correspondence between
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Eidsath, A., Carbonell, R.G., Whitaker, S., Herrmann, L.R., 1983. Dispersion in
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Fried, J.J., Combarnous, M.A., 1971. Dispersion in porous media. Advances in
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Gerke, H.H., Genuchten, M.V., 1993. A dual-porosity model for simulating the
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re
Gunn, D.J., Pryce, C., 1969. Dispersion in porous media. Trans. Chem. Eng. 47,
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lP
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9
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