MASS TRANSFER IN
MULTIPHASE SYSTEMS
AND ITS APPLICATIONS
Edited by Mohamed El-Amin
Mass Transfer in Multiphase Systems and its Applications
Edited by Mohamed El-Amin
Published by InTech
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Copyright © 2011 InTech
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Mass Transfer in Multiphase Systems and its Applications, Edited by Mohamed El-Amin

Anna Kiełbus-Rąpała and Joanna Karcz
Gas-Liquid Mass Transfer in an Unbaffled
Vessel Agitated by Unsteadily Forward-Reverse
Rotating Multiple Impellers 117
Masanori Yoshida, Kazuaki Yamagiwa,
Akira Ohkawa and Shuichi Tezura
Toward a Multiphase Local Approach in the
Modeling of Flotation and Mass Transfer
in Gas-Liquid Contacting Systems 137
Jamel Chahed and Kamel M’Rabet
Mass Transfer in Two-Phase Gas-Liquid Flow
in a Tube and in Channels of Complex Configuration 155
Nikolay Pecherkin and Vladimir Chekhovich
Contents
Contents
VI
Laminar Mixed Convection Heat and Mass Transfer
with Phase Change and Flow Reversal in Channels 179
Brahim Benhamou, Othmane Oulaid,
Mohamed Aboudou Kassim and Nicolas Galanis
Liquid-Liquid Extraction
With and Without a Chemical Reaction 207
Claudia Irina Koncsag and Alina Barbulescu
Modeling Enhanced Diffusion Mass
Transfer in Metals during Mechanical Alloying 233
Boris B. Khina and Grigoriy F. Lovshenko
Mass Transfer in Steelmaking Operations 255
Roberto Parreiras Tavares
Effects of Surface Tension on Mass Transfer Devices 273
Honda (Hung-Ta) Wu and Tsair-Wang Chung

Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Chapter 19
Contents
VII
Condensation Capture of Fine Dust in Jet Scrubbers 459
M.I. Shilyaev and E.M. Khromova
Mass Transfer in Filtration Combustion Processes 483
David Lempert, Sergei Glazov and Georgy Manelis
Mass Transfer in Hollow Fiber Supported Liquid Membrane
for As and Hg Removal from Produced Water in Upstream
Petroleum Operation in the Gulf of Thailand 499
U. Pancharoen, A.W. Lothongkum and S. Chaturabul
Mass Transfer in Fluidized
Bed Drying of Moist Particulate 525
Yassir T. Makkawi and Raffaella Ocone
Simulation Studies on the Coupling Process
of Heat/Mass Transfer in a Metal Hydride Reactor 549
Fusheng Yang and Zaoxiao Zhang
Mass Transfer around Active Particles in Fluidized Beds 571
Fabrizio Scala
Mass Transfer Phenomena and Biological Membranes 593
Parvin Zakeri-Milani and Hadi Valizadeh
Heat and Mass Transfer
in Packed Bed Drying of Shrinking Particles 621

tems that are usually encountered in many research areas such as chemical, reactor,
environmental and petroleum engineering. From biological and chemical reactors to
paper and wood industry and all the way to thin fi lm, the 31 chapters of this book
serve as an important reference for any researcher or engineer working in the fi eld of
mass transfer and related topics.
The fi rst chapter focuses on the description and modeling of mass transfer processes
occurring between two fl uid phases in a porous medium, while the second chapter is
concerned with the basic principles underlying transport phenomena and chemical
reaction in single- and multi-phase systems in porous media. Chapter 3 introduces the
multiphase modeling of thermomechanical behavior of early-age silicate composites.
The surfactant transfer in multiphase liquid systems under conditions of weak gravita-
tional convection is presented in Chapter 4.
In the fi  h chapter the volumetric mass transfer coeffi cient for multiphase mechani-
cally agitated gas–liquid and gas–solid–liquid systems is obtained experimentally.
Further, gas-liquid mass transfer analysis in an unbaffl ed vessel agitated by unsteadily
forward-reverse rotating multiple impellers is provided in Chapter 6. Chapter 7 dis-
cuses the kinetic model of fl otation based on the theory of mass transfer in gas-liquid
bubbly fl ows. The eighth chapter deals with experimental investigation of mass trans-
fer and wall shear stress, and their interaction at the concurrent gas-liquid fl ow in a
vertical tube, in a channel with fl ow turn, and in a channel with abrupt expansion. The
laminar mixed convection with mass transfer and phase change of fl ow reversal in
channels is studied in the ninth chapter, and the tenth chapter exemplifi es the theoreti-
cal aspects of the liquid-liquid extraction with and without a chemical reaction and the
dimensioning of the extractors with original experimental work and interpretations.
The eleventh chapter introduces analysis of the existing theories and concepts of solid-
state diff usion mass transfer in metals during mechanical alloying. In Chapter 12 the
mass transfer coeffi cient is given for diff erent situations (liquid-liquid, liquid-gas and
liquid-solid) of two-phase mass transfer of steelmaking processes. Chapter 13 discuss-
es the eff ect of Marangoni Instability on thin liquid fi lm, thinker liquid layer and mass
transfer devices.

The objective of the twenty-seventh chapter is to provide comprehensive information
on theoretical-experimental analysis of coupled heat and mass transfer in packed bed
drying of shrinking particles. The twenty-eighth chapter focuses on vapour-liquid
mass transfer infl uence on the prediction of RD column behaviour neglecting the liq-
uid-solid and intraparticle mass transfer. Mass transfer through catalytic membrane
layer is studied in Chapter 29. In chapter 30 three types of bioreactors and stirred tank
applied to biological systems are introduced and a mathematical model is developed.
Finally in Chapter 31 analytical solutions of mass transfer around a prolate or an oblate
spheroid immersed in a packed bed are obtained.
Mohamed Fathy El-Amin
Physical Sciences and Engineering Division
King Abdullah University of Science and Technology (KAUST)


0
Mass and Heat Transfer During Two-Phase Flow in
Porous Media - Theory and Modeling
Jennifer Niessner
1
and S. Majid Hassanizadeh
2
1
Institute of Hydraulic Engineering, University of Stuttgart, Stuttgart
2
Department of Earth Sciences, Faculty of Geosciences, Utrecht University, Utrecht
1
Germany
2
The Netherlands
1. Introduction

www.oxy.com)
groundwater
precipitation
radiation
evaporation
infiltration
(d) Evaporation from soil
Fig. 2. Four applications of flow and transport in porous media where interphase mass
transfer is important
2
Mass Transfer in Multiphase Systems and its Applications
Mass and Heat Transfer During Two-Phase Flow in Porous Media - Theor y and Modeling 3
(a) Carbon capture and storage (Fig. 2 (a)) is a recent strategy to mitigate the greenhouse
effect by capturing the greenhouse gas carbon dioxide that is emitted e.g. by coal power
plants and inject it directly into the subsurface below an impermeable caprock. Here, three
different storage mechanisms are relevant on different time scales: 1) The capillary barrier
mechanism of the caprock. This geologic layer is meant to keep the carbon dioxide in the
storage reservoir as a separate phase. 2) Dissolution of the carbon dioxide in the surrounding
brine (salty groundwater). This is a longterm storage mechanism and involves a mass
transfer process as carbon dioxide molecules are “transferred” from the gaseous phase to
the brine phase. 3) Geochemical reactions which immobilize the carbon dioxide through
incorporation into the rock matrix.
(b) Shown in Fig. 2 (b) is a cartoon of a light non-aqueous phase liquid (LNAPL)
soil contamination and its clean up by injection of steam at wells located around the
contaminated soil. The idea behind this strategy is to mobilize the initially immobile
(residual) LNAPL by evaporation of LNAPL component at large rates into the gaseous
phase. The soil gas is then extracted by a centrally located extraction well. It means that the
remediation mechanism relies on the evaporation of LNAPL component which represents a
mass transfer from the liquid LNAPL phase into the gaseous phase.
(c) In order to produce an additional 8-20% of oil after primary and secondary recovery,

solid
phase
w
wetting
fluid phase
n
non−wetting
fluid phase
pore scale
macro scale
Fig. 3. Pore-scale versus macro-scale description of flow and transport in a porous medium.
scale. From there, we try to get a better understanding of the macro-scale physics of mass
transfer, which is our scale of interest.
In Fig. 1 we have seen that interphase mass transfer is inherently a pore-scale process as
it—naturally—takes place across fluid–fluid interfaces. Let us imagine a situation where two
fluid phases, a wetting phase and a non-wetting phase, are brought in contact as shown in
Fig. 4. Commonly, when the two phases are brought in contact (time t
= t
0
), equilibrium is
quickly established directly at the interface. With respect to mass transfer, this means that
the concentration of non-wetting phase particles in the wetting phase at the interface as well
as the concentration of wetting-phase particles in the non-wettting phase at the interface are
both at their equilibrium values, C
2
1,eq
and C
1
2,eq
.Att = t

C
2
1,eq
1
2,eq
C
x
t = t
0
solid phase
fluid
phase 1
fluid phase 2
C
x
t = t
1
solid phase
fluid
phase 1
fluid phase 2
C
x
t = t
2
C
2
1,eq
Fig. 4. Pore-scale picture of interphase mass transfer.
4

is small compared to that of flow. However, if large flow velocities occur as e.g. during air
sparging, the local equilibrium assumption gives completely wrong results, see Falta (2000;
2003) and van Antwerp et al. (2008). We will investigate and quantify this issue later in Sec. 3.3
.
Local equilibrium models for multi-phase systems have been introduced and developed
e.g. by Miller et al. (1990); Powers et al. (1992; 1994); Imhoff et al. (1994); Zhang & Schwartz
(2000) and have been used and advanced ever since. Let us consider a system with a liquid
phase (denoted by subscript l) and a gaseous phase (denoted by subscript g) composed of
air and water components. Then, Henry’s Law is employed to determine the mole fraction
of air in the liquid phase, while the mole fraction of water in the gas phase is determined by
assuming that the vapor pressure in the gas phase is equal to the saturation vapor pressure.
Denoting the water component by superscript w and the air component by superscript a, this
yields
x
a
l
= p
a
g
· H
a
l
−g
(1)
x
w
g
=
p
w

in the gas phase while p
g
[Pa] is the gas pressure. The remaining mole fractions result simply
from the condition that mole fractions in each phase have to sum up to one,
x
w
l
= 1 − x
a
l
(3)
x
a
g
= 1 − x
w
g
. (4)
Note that while for a number of applications the equilibrium mole fractions are constants or
merely a function of temperature, in our case, they will be functions of space and time as
pressure and the composition of the phases changes.
2.2.2 Classical kinetic approach
Kinetic mass transfer approaches are traditionally applied to the dissolution of contaminants
in the subsurface which form a separate phase from water, the so-called non-aqueous phase
liquids (NAPLs). If such a non-aqueous phase liquid is heavier than water, it is called
“dense non-aqueous phase liquid” or DNAPL. When an immobile lense of DNAPL is present
at residual saturation (i.e. at a saturation which is so low that the phase is immobile) and
dissolves into the surrounding groundwater, the kinetics of this mass transfer process usually
plays an important role: the dissolution of DNAPL is a rate-limited process. This is also
the case when a pool of DNAPL is formed on an impermeable layer. In these relatively

3
s

is the interphase mass transfer rate of component κ from phase α to
phase β, k
κ
α
→β

m
s

is the mass transfer rate coefficient, a
αβ

1
m

is the specific interfacial
area separating phases α and β, C
κ
β,s

kg
m
3

is the solubility limit of component κ in phase
β, and finally, C
κ

Schwartz (2000)). This yields, in a simplified notation,
Q
= k(C
s
−C). (6)
Here, C
s
is the solubility limit of the DNAPL component in water and C is its actual
concentration. The lumped mass transfer coefficient k

1
s

is commonly related to a modified
Sherwood number Sh by
k
= Sh
D
m
d
2
50
, (7)
6
Mass Transfer in Multiphase Systems and its Applications
Mass and Heat Transfer During Two-Phase Flow in Porous Media - Theor y and Modeling 7
where D
m

m

3.1 Theoretical background
Due to a number of deficiencies of the classical model for two-phase flow in porous media
(one of which is the problem in describing kinetic interphase mass transfer on the macro
scale), several approaches have been developed to describe two-phase flow in an alternative
and thermodynamically-based way. Among these are a rational thermodynamics approach
by Hassanizadeh & Gray (1980; 1990; 1993b;a), a thermodynamically constrained averaging
theory approach by Gray and Miller (e.g. Gray & Miller (2005); Jackson et al. (2009)),
mixture theory (Bowen (1982)) and an approach based on averaging and non-equilibrium
thermodynamics by Marle (1981) and Kalaydjian (1987). While Marle (1981) and Kalaydjian
(1987) developed their set of constitutive relationships phenomenologically, Hassanizadeh &
Gray (1990; 1993b); Jackson et al. (2009), and Bowen (1982) exploited the entropy inequality
to obtain constitutive relationships. To the best of our knowledge, the two-phase flow models
of Marle (1981); Kalaydjian (1987); Hassanizadeh & Gray (1990; 1993b); Jackson et al. (2009)
are the only ones to include interfaces explicitly in their formulation allowing to describe
hysteresis as well as kinetic interphase mass and energy transfer in a physically-based way. In
the following, we follow the approach of Hassanizadeh & Gray (1990; 1993b) as it includes
the spatial and temporal evolution of phase-interfacial areas as parameters which allows
us to model kinetic interphase mass transfer in a much more physically-based way than is
classically done.
It has been conjectured by Hassanizadeh & Gray (1990; 1993b) that problems of the classical
two-phase flow model, like the hysteretic behavior of the constitutive relationship between
capillary pressure and saturation, are due to the absence of interfacial areas in the theory.
Hassanizadeh and Gray showed (Hassanizadeh & Gray (1990; 1993b)) that by formulating
the conservation equations not only for the bulk phases, but additionally for interfaces,
and by exploiting the residual entropy inequality, a relationship between capillary pressure,
saturation, and specific interfacial areas (interfacial area per volume of REV) can be derived.
This relationship has been determined in various experimental works (Brusseau et al. (1997);
Chen & Kibbey (2006); Culligan et al. (2004); Schaefer et al. (2000); Wildenschild et al. (2002);
7
Mass and Heat Transfer During Two-Phase Flow in Porous Media - Theory and Modeling

= w, a):
∂

φS
l
¯
ρ
l
¯
X
κ
l

∂t
+ ∇·
(
φS
l
¯
ρ
l
¯
X
κ
l
¯
v
l
)
−∇·

−v
l

+ j
κ
l

·n
lg
dA (9)
∂

φS
g
¯
ρ
g
¯
X
κ
g

∂t
+ ∇·

φS
g
¯
ρ
g


ρ
g
X
κ
g

v
g
−v
lg

− j
κ
g

·n
lg
dA (10)
mass balance for lg-interface components (κ
= w, a):
∂

¯
Γ
lg
¯
X
κ
lg

1
V

A
lg

ρ
l
X
κ
l

v
l
−v
lg

− j
κ
l
−ρ
g
X
κ
g

v
g
−v
lg

l
¯
v
l
¯
v
l
)
−∇·
(
φS
l
T
l
)
=
1
V

A
lg

ρ
l
v
l

v
lg
−v

¯
v
g

−∇·

φS
g
T
g

=
1
V

A
lg

ρ
g
v
g

v
g
−v
lg

−t
g


−∇·

T
lg
a
lg

=
1
V

A
lg

ρ
l
v
l

v
l
−v
lg

−t
l
−ρ
g
v

kg·m
4
s

is the diffusive flux of component κ in phase α, V is the magnitude of
the averaging volume, A
lg
denotes the interfaces separating the l-phase and the g-phase in an
averaging volume, v
lg

m
s

is the velocity of the lg-interface, and n
lg
is the unit vector normal
to A
lg
and pointing into the g-phase. Furthermore, X
κ
lg
[−] is the mass fraction of component κ
in the lg-interface, j
κ
lg

kg·m
4
s

micro-scale diffusive fluxes j
κ
α
to the local concentration gradient resulting in the following
approximation:
j
κ
α
·n
lg
= ±
ρ
α
D
κ
d
κ
a
lg

X
κ
α,s
− X
κ
α

, (15)
where D
κ

¯
X
w
l

∂t
+ ∇·
(
¯
ρ
l
¯
X
w
l
¯
v
l
)
−∇·

¯
j
w
l

= ρ
l
Q
w

l

∂t
+ ∇·
(
¯
ρ
l
¯
X
a
l
¯
v
l
)
−∇·

¯
j
a
l

= ρ
l
Q
a
l
+
D

+ ∇·

¯
ρ
g
¯
X
w
g
¯
v
g

−∇·

¯
j
w
g

= ρ
g
Q
w
g
+
D
w
¯
ρ

ρ
g
¯
X
a
g
¯
v
g

−∇·

¯
j
a
g

= ρ
g
Q
a
g
−
D
a
¯
ρ
l
d
a

l
μ
l

∇p
l
−
¯
ρ
l
g

(21)
¯
v
g
= −K
S
2
g
μ
g

∇p
g
−
¯
ρ
g
g

w
g
+ X
a
g
= 1 (27)
a
lg
= a
lg
(S
l
, p
c
), (28)
The macro-scale mass fluxes
¯
j
κ
α
are given by a Fickian dispersion equation,
¯
j
κ
α
= −ρ
α
¯
D
κ

Depending on the parameters, initial conditions, and boundary conditions of the system,
kinetics might be important for mass transfer. If so, then it may not be sufficient to use a
classical local equilibrium model instead of the more complex interfacial-area-based model.
To allow for a decision, Niessner & Hassanizadeh (2009a) make the system of equations (16)
through (28) dimensionless and study the dependence of kinetics on Damk
¨
ohler number and
Peclet number.
To do so, they define dimensionless variables:
t
∗
=
tv
R
φL
,
∇
∗
= L∇,
¯
v
∗
α
=
¯
v
α
v
R
, Q

α
D
R,α
, v
∗
lg
=
φ
v
R
v
lg
, (31)
E
∗
lg
=
a
R,lg
φL
v
R
E
lg
, g
∗
α
=
¯
ρ

R
, ρ
∗
g
=
¯
ρ
g
¯
ρ
l
μ
∗
l
=
μ
l
μ
g
. (33)
Here, ρ
∗
g
is density ratio, μ
∗
l
is viscosity ratio, v
R
is a reference velocity, L is a characteristic
length, a

R
L
D
R,α
,Da
κ
=
D
κ
La
R,lg
dv
R
. (34)
These definitions lead to the following dimensionless form of Eq. (16) through (28):
∂
∂t
∗
(
S
l
¯
X
w
l
)
+ ∇
∗
·
(

∗
lg
ρ
∗
g

X
w
g,s
−
¯
X
w
g

(35)
∂
∂t
∗
(
S
l
¯
X
a
l
)
+ ∇
∗
·

a
∗
lg

X
a
l,s
−
¯
X
a
l

(36)
∂
∂t
∗

S
g
¯
X
w
g

+ ∇
∗
·

¯

lg

X
w
g,s
−
¯
X
w
g

(37)
∂
∂t
∗

S
g
¯
X
a
g

+ ∇
∗
·

¯
X
a

ρ
∗
g

X
a
l,s
−
¯
X
a
l

(38)
∂a
∗
lg
∂t
∗
+ ∇
∗
·

a
∗
lg
v
∗
lg



∇
∗
p
∗
g
− g
∗
g

(41)
11
Mass and Heat Transfer During Two-Phase Flow in Porous Media - Theory and Modeling
12 Mass Transfer
v
∗
lg
= −K
∗
lg
∇
∗
a
∗
lg
(42)
p
∗
c
= p

= 1 (46)
a
∗
lg
= a
∗
lg
(
S
l
, p
∗
c
)
. (47)
In order to investigate the importance of kinetics, we define Pe :
= Pe
l
= Pe
g
and Da := Da
w
=
Da
a
and vary Pe and Da independently over five orders of magnitude. Therefore, we consider
a numerical example where dry air is injected into a horizontal (two-dimensional) porous
medium of size 0.7 m
× 0.5 m that is almost saturated with water (initial and boundary water
saturation of 0.9).

0.014
0m 0.7m
4.0E-5
0m
0.7m
0.014
0m 0.7m
4.0E-5
0m
0.7m
4.0E-5
0m
0.7m
4.0E-5
0m
0.7m
4.0E-5
0m
0.7m
0.014
0m 0.7m
4.0E-5
0m
0.7m
0.014
0m 0.7m
0.014
0m 0.7m
0.014
0m 0.7m

xxxx
xxxx
xxxx
0
0.7
0
0
0
0.7
0
0
0.7
0
0
0.7
0.70
0
0.0140.0140.014 0.014 0.014
0
0
0.7
0.70
00
0 0.7
0
0 0.7
0
0 0.7
0
0

0
0.7
4E−5 4E−5 4E−5 4E−5 4E−5
X [−]
4E−5 4E−5 4E−5 4E−5 4E−5
X [−]
X [−]
Fig. 5. Solubility limits X
a
l,s
and X
w
g,s
and actual mass fractions
¯
X
a
l
and
¯
X
w
g
for two different
time steps (0.0035 s and 0.01 s) and 5 different Damk
¨
ohler numbers.
12
Mass Transfer in Multiphase Systems and its Applications
Mass and Heat Transfer During Two-Phase Flow in Porous Media - Theor y and Modeling 13

that kinetic interphase mass transfer is independent of Peclet number, at least within the four
orders of magnitude considered here.
4. Extension to heat transfer
The concept of describing mass transfer based on modeling the evolution of interfacial areas
using the thermodynamically-based approach of Hassanizadeh & Gray (1990; 1993b) can
be extended to describing interphase heat transfer as well. The main difference between
interphase mass and heat transfer is that, in addition to fluid–fluid interfaces, heat can also be
transferred across fluid–solid interfaces, see Fig. 6.
Similarly to mass transfer, classical two-phase flow models describe heat transfer on the
macro scale by either assuming local thermal equilibrium within an averaging volume or
by formulating empirical models to describe the transfer rates. The latter is necessary
as classically, both fluid–fluid and fluid–solid interfacial areas are unknown on the macro
scale. And similiarly to mass transfer, we can use the thermodynamically-based approach
of Hassanizadeh & Gray (1990; 1993b) which includes both fluid–fluid and fluid–solid
interfacial areas in order to describe mass transfer in a physically-based way. We can
non−wetting
wetting
(1)
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ass

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non−wetting