Jordan Journal of Civil Engineering, Volume 1, No. 3, 2007
Perturbation Solution of the Jackson’s Dusty Gas Model Equations
for Ternary Gaseous Systems
Wa’il Y. Abu-El-Sha’r1), Riyadh Al-Raoush2) and Khaled Asfar3)
1)
Associate Professor, Department of Civil Engineering, Jordan University of Science and Technology, P.O. Box 3030,
Irbid -22110, Jordan, Fax: (962)27095018, E-mail: [email protected]. (Corresponding Author).
2)
Assistant Professor, Department of Civil and Environmental Engineering, Southern University and A&M College,
Baton Rouge, LA 70813. E-mail: [email protected].
3)
Professor, Department of Mechanical Engineering, Jordan University of Science and Technology, Irbid, Jordan.
ABSTRACT
A perturbation technique was used to obtain an approximate closed-form solution for the mass balance equations
when the dusty gas model (DGM) is used to calculate total molar fluxes of components of ternary gaseous
systems. This technique employed the straight-forward expansion method to the second-order approximation.
Steady-state, isobaric, isothermal and no reaction conditions were assumed. The obtained solution is a set of
equations expressed to calculate mole fractions as functions of dimensionless length, boundary conditions,
properties of the gases and parameters of transport mechanisms (i.e., Knudsen diffusivity and effective binary
diffusivity). Three different systems represent field and experimental conditions were used to test the
applicability of perturbation solution. Findings indicate that the obtained solution provides an effective tool to
calculate mole fractions and total molar fluxes of components of ternary gaseous systems.
KEYWORDS: Dusty Gas Model, Perturbation, Gas Transport Mechanisms, Diffusive Transport,
Porous Media, VOC, Groundwater Contamination.
INTRODUCTION
Many significant environmental problems require
quantification of diffusive transport mechanisms in the
gaseous phase. Examples of such problems include: the
use of natural unsaturated zones as landfills and disposal
sites for hazardous wastes, groundwater contamination
by volatile organic compounds (VOCs), subsurface
remediation and the health effects of Radon and its
decay products.
Three models have been commonly used to model
Received on 15/5/2007 and Accepted for Publication on
1/7/2007.
diffusive transport in natural porous media. These
include: Fick’s first law of diffusion, the Stefan-Maxwell
equations and the Dusty Gas Model (DGM). Many
studies investigated the applicability of Fick’s first law of
diffusion to model vapor diffusion and highlighted the
importance of flux mechanisms other than molecular
diffusion flux to adequately model the diffusive transport
in a porous medium. Thorestenson and Pollock (1989a, b)
used the Stefan-Maxwell equations and the DGM to
assess the limitations of Fick’s fist law through a
theoretical investigation of the relative importance of
different transport mechanisms in binary and multicomponent gaseous systems. Baehr and Bruell (1990)
analyzed results of hydrocarbon vapor transports column
experiments and calculated the tortuosity factors
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© 2007 JUST. All Rights Reserved.
Perturbation Solution…
Wa'il Y. Abu-El-Sha'r, Riyadh Al-Raoush and Khaled Asfar
necessary to fit these experimental data when using
Fick’s law and the Stefan-Maxwell equations. Abriola et
al. (1990) numerically investigated the importance of
different gas transport mechanisms in non-steady-state
binary systems. Voudrias and Li (1992) experimentally
investigated the importance of Knudesn diffusion in an
unsaturated soil sample containing benzene vapor. AbuEl-Sha’r and Abriola (1997) experimentally evaluated the
relative importance of different gaseous transport
mechanisms in natural porous media systems.
Findings of the above studies indicated that a
multicomponent treatment incorporating different gas
transport mechanisms should be undertaken to obtain
better modeling of gas transport in natural porous
media. This is of special importance when modeling
systems where Knusden diffusion contributes
significantly to the total diffusion (e.g., clay). In such
systems, Fick’s diffusion and the Stefan-Maxwell
equations may be inadequate to model diffusive gaseous
transport. Therefore, analysis of natural environmental
problems requires consideration of ternary systems
(systems consisting of a gaseous contaminate and air
which is usually modeled as a mixture of oxygen and
nitrogen).
The use of the DGM to solve for concentrations and
molar fluxes of species of gaseous systems when the
number of components of three or more exists is limited
by the availability of a closed-form solution to the DGM.
To our knowledge, there is no analytical solution in the
literature to the mass balance equations when the DGM is
used to calculate concentrations or total molar fluxes of
gaseous species in a porous medium. Numerical solutions
however are limited and have their own limitations. The
objectives of this paper are: (1) to obtain an approximate
closed-form solution to the DGM equations when
incorporated in mass balance equations for ternary
systems using perturbation methods. The following
conditions are also assumed: steady-state, isothermal, no
chemical or biological reactions occur in the system and
surface diffusion is neglected, (2) to apply the obtained
approximate closed-form solution to different natural
porous media systems.
- 246 -
BACKGROUND
Gas Transport Mechanisms through Porous Media
Gas transport through a porous medium occurs via
four different mechanisms: (1) surface flow or diffusion;
(2) viscous flow; (3) Knudsen flow; (4) ordinary
diffusion (i.e., molecular and non-equimolar fluxes). A
brief discussion of these mechanisms is given below and
a detailed discussion can be found in Cunningham and
Williams (1980), Mason and Malinauskas (1983) and
Abu-El-Sha’r (1993).
Surface flux occurs when gas molecules are adsorbed
on specific sites at the surface of the particles of the
porous media. Due to the continuous movement
(vibrations) of the adsorbed molecules, each molecule
transfers by hopping to other adsorption sites a number of
times before it returns to the gaseous phase. Surface
diffusion is usually modeled by employing Fick’s Law of
diffusion where the concentration gradients refer to the
surface concentration gradients and all the complexities
of the porous medium geometry, surface structure and
adsorption equilibrium are lumped into the surface
diffusion coefficient. The Fickian model is useful only at
low surface coverage (Mason and Malinuskas, 1983). In
this paper, the number of molecules adsorbed to the soil
surface adsorption sites is assumed equal to the number
of molecules leaving the adsorption sites (steady-state
conditions). Thus, net surface flux will not effect the total
gaseous flux and is neglected.
Viscous flux occurs when a pressure gradient is
applied on the system. The damping effects due to the
high rate of interaction (i.e., collisions) among gas
molecules compared to the interaction between gas
molecules and the boundaries of the system cause a
constant viscous flux. On the other hand, when there is
more interaction between gas molecules and system
boundaries than the interaction among gas molecules,
Knudsen flux dominates. In a multicomponent gaseous
system, there is a concentration gradient for each
component and thus a net Knudsen flux of each
component as well. The net Knudsen flux of gas i can be
calculated as (Cunningham and Williams, 1980):
Jordan Journal of Civil Engineering, Volume 1, No. 3, 2007
N iK = − DiK
dci
dz
(1)
where N iK is the Knudsen molar flux of component i ,
ci is molar concentration of component i and DK is the
i
Knudsen diffusivity, given as:
⎛ RT ⎞
⎟⎟
D = Q p ⎜⎜
M
⎝ i⎠
1/ 2
K
i
(2)
where Q p is the obstruction factor for Knudsen
diffusivity, T is the temperature, R is the ideal gas
constant and M i is the molecular weight of gas i . For a
K
porous media with a single pore size, Di is given by
(Geankoplis, 1972):
3
DiK = 9.7 × 10 r
T
Mi
(N )
D
i F
= − Dij c
dX i
dz
(6)
( )
D
where N i F is Fick’s first law diffusive molar flux
of component i , Dij is the binary diffusivity of
components i and j and c is the total molar
concentration. Dij can be calculated as (Perry and Green,
1997):
(3)
where r is the average pore radius.
Ordinary diffusion of gases in a porous medium is a
combination of two different flux mechanisms: diffusive
(molecular) flux and viscous (nonequimolar) flux. The
total molar diffusive flux of component i is given by
(Cunningham and Williams, 1980):
N iT = N iD + X i N v
porous media. Fick’s law has been reported in the
literature of hydrology and soil physics to model
molecular diffusion. It originated based on solute studies
as an empirical equation and then extended for prediction
of gaseous diffusion through porous media (Kirham and
Powers, 1972; Abu-El-Sha’r, 1993).
Fick’s law for one-dimensional and steady-state
conditions is written in molar form as (Jaynes and
Rogwski, 1983):
−3
10 T
Dij =
P
1.75
⎡( M i + M j ) ⎤
⎢
⎥
⎣⎢ M i M j ⎥⎦
1/ 2
[(∑V ) + (∑V ) ]
1/ 3 2
1/ 3
i
(7)
i
here T is the temperature (Kelvin), P is the pressure
(Kpa), and ∑ V is the sum of atomic diffusion volumes.
For gaseous transport through porous media, Dij is
replaced by the effective diffusive coefficient Dije , which
(4)
is given by:
where N iT is the total diffusive flux of component i ,
N iD is molar diffusive flux of component i , X i is the
v
mole fraction of component i and N i is the molar
viscous flux of component and can be given as:
Nv = −
P k
∇P
RT µ
where k is the intrinsic permeability, µ
dynamic viscosity and ∇P is the pressure gradient.
(5)
is the
Gas Transport Models
As mentioned previously, Fick’s first law of diffusion,
Stefan-Maxwell equations and the Dusty Gas Model
(DGM) have been used to study gas transport through
Dije = Qm Dij
(8)
where Qm is an obstruction factor, which is a function
of porosity and tortuosity of the porous medium. One
suggested correlation for Qm is given by (Cunningham
and Williams, 1980):
Q m = TP 3
(9)
where TP is the porosity of the porous media.
Stefan-Maxwell equations are usually used in
chemical engineering to study multicomponent gas
diffusion. The general form of Stefan-Maxwell equations
can be given as:
- 247 -
Perturbation Solution…
X i N jD − X j N iD
n
∑
=
e
ij
D
j =1 , j ≠i
Wa'il Y. Abu-El-Sha'r, Riyadh Al-Raoush and Khaled Asfar
1
∇Pi
RT
(10)
where n is the number of gaseous component in the
system. Note that Equation (10) represents n − 1
independent equations, since
n
∑
X
i
= 1 .0
(11)
i=1
The form of Stefan-Maxwell equations given in
Equation (10) can be written in a form similar to Fick’s
law, where diffusion coefficients are function of the
composition of gaseous components and properties of the
porous media. It is to be noted that both Fick’s law of
diffusion and Stefan-Maxwell equations do not
incorporate Knudsen diffusion, therefore, both models
may be inadequate to model systems where Knudsen
diffusion is significant.
Although the DGM can be used to model
multicomponent gaseous transport through porous
media, it has been rarely used in natural systems
n
X i N jD − X j N iD
j =1 , j ≠ i
Dije
∑
−
applications (Alzyadi, 1975; Thorstenson and Pollock,
1989a, b; Abriola et al., 1992; Voudrias and Li, 1992).
In addition, few measurements have been made to
estimate the different transport parameters and
coefficients incorporated in the model (Allawi and
Gunn, 1987; Abu-El-Sha’r and Abriola, 1997). The
DGM incorporates the different transport mechanisms
(i.e., molecular diffusion, non-equimolar flux, Knudsen
diffusion, surface diffusion and viscous flux) in a
rigorous way based on the kinetic theory treatment. The
DGM treats the porous medium as a collection of
suspended large dust particles. The dust particles (i.e.,
solid matrix) are considered as one component of the
gaseous mixture, uniformly distributed, and much larger
and heavier than the gas molecules (Jackson, 1977;
Cunningham and Williams, 1980; Mason and
Malinauslcas, 1983).
The constitutive equations of the DGM in molar form
are given by:
n
N iD
1
=
∇
P
+
n
'
X i X j α ij ∇ ln T
i
∑
DiK
RT
j =1
where n' is the gas and particle density, α ij is the
generalized thermal diffusivity and the remaining
parameters are previously defined. The summation term
on the left hand side of Equation (12) is the momentum
lost through molecule-molecule collisions with
component other than i (but not the particle), the
second term is the momentum lost by component i
through molecule-particle collisions, the first term on
the right hand side is the component pressure gradient
of component i and the second term is the thermal
gradient which may be neglected for isothermal
systems.
Analysis of Gaseous Systems
The analysis of a particular gaseous system requires
solution of mass balance equations for system
components based on given boundary and initial
conditions. The mass balance equations on a molar basis,
assuming uniform void fraction in time, can be written in
- 248 -
(12)
a general form as:
∇ .N iT + ε
∂c i
+ R vi = 0
∂t
(13)
where N iT is the total molar flux of component i
which can be calculated using either Fick’s law, StefanMaxwell equations, or DGM; ε is the void fraction of
the porous medium; ci is the number of moles per unit
void volume; Rvi is the reaction rate of species i per unit
volume of porous media.
In this paper, the following conditions were assumed:
isothermal, steady-state, isobaric, no chemical or
biological reactions occur in the system and surface
diffusion is insignificant. Therefore, Equation (13) is
reduced to:
∇.N iT = 0
(14)
For ternary gas system, the total molar fluxes for the
different gas components may be explicitly written using
the DGM equation as (Feng and Stewart, 1973):
Jordan Journal of Civil Engineering, Volume 1, No. 3, 2007
2
[N z ] = −[c] r dp − [F ( r )]−1 ⎡⎢ dc ⎤⎥ − [Ds ( r )]⎡⎢ dc ⎤⎥
8 µ dz
⎣ dz ⎦
⎣ dz ⎦
(15)
where [N z ] and [c ] are n-element column vectors
consisting of elements Niz and ci; respectively; r is the
radius of pore; µ is the gas dynamic viscosity; [Ds ( r )] is
an n×n diagonal matrix of surface diffusivities, Dis ( r ) ;
[F ( r )] is the n×n matrix formed as:
Fij (r ) = −
Fij (r ) =
Xi
, (i ≠ j )
Dij
n
X
1
+ ∑ h
D (r ) h =1, h ≠ i Dih
K
i
(16-a)
(16-b)
For steady-state, isobaric conditions, Equation 15 may
be written as:
[N z ] = −[F ( r )]−1 ⎡⎢ dc ⎤⎥
(17)
⎣ dz ⎦
Perturbation
Perturbation theory is a collection of methods for
systematic analysis of the global behavior of solutions to
the differential equations. The general procedure of
perturbation theory is to identify a parameter (small or
large), usually denoted by∈, then the solution is
represented by the first few terms of an asymptotic
expansion, usually not more than two terms. The
expansion may be carried out in terms of ∈ which
appears naturally in the equations, or may be artificially
and temporarily introduced into a difficult problem
X2
X3
⎡ 1
⎡ N1 ⎤
⎢ DK + D + D
⎢ ⎥
12
13
⎢ 1
⎢ ⎥
⎢
⎢ ⎥
⎢
X
⎢N ⎥
− 2
⎢
⎢ 2 ⎥ = −⎢
D21
⎢ ⎥
⎢
⎢ ⎥
⎢
X
⎢ ⎥
⎢
− 3
N
⎢ 3⎥
⎢
D31
⎢ ⎥
⎢
⎣ ⎦
⎣
−
having no small parameter. Then ∈=1 is set, if necessary,
to recover the original problem. This artificial conversion
to a perturbation problem may be the only way to solve
the problem. However, it is preferable to introduce ∈ in
such away that the zero-order solution (i.e., the leading
terms in the perturbation series) is obtainable as a closedform analytic expression. Such expansion is called
parameter perturbation. Alternatively, the expansion may
be carried out in terms of a coordinate (either small or
large). These are called coordinate perturbation. The idea
of perturbation theory is to decompose a tough problem
into an infinite number of relatively easy ones. Hence,
perturbation theory is most useful when the first few
terms reveal the important features of the solution and the
remaining ones give small corrections (Bender and
Orszag, 1999; Nayfeh, 2000).
Details of Solution of the DGM
Molar fluxes of species along a given direction (i.e., the
z direction) in a multicomponent gaseous system are
given by Equation (15). For steady-state diffusive mass
transport, it is commonly assumed that viscous flow and
surface diffusion are insignificant (Farmer et al., 1980;
Karimi et al., 1987; Baeher and Bruell, 1990; Voudrious
and Li, 1992). Therefore, the first and third terms of the
right hand side of Equation (15) may be neglected.
Using Equations (15) and (16) and the relations: Pi=Xi P
and Ci = Pi RT , the molar fluxes of a three-component
gaseous system can be given as (i.e., Equation 17):
⎤
⎥
⎥
⎥
⎥
X2
−
⎥
D23
⎥
⎥
⎥
1
X1
X2 ⎥
+
+
D3K D31 D32 ⎥
⎥
⎦
X1
D12
−
1
X
X
+ 1 + 3
K
D2
D21 D23
−
X3
D32
X1
D13
-1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣⎢
P •
X1
RT
P •
X2
RT
P •
X3
RT
⎤
⎥
⎥
⎥
⎥
⎥ (18)
⎥
⎥
⎥
⎥
⎥
⎦⎥
Dots in the above equation represent the derivative
with respect to z. For formulation convenience, let:
- 249 -
Perturbation Solution…
Wa'il Y. Abu-El-Sha'r, Riyadh Al-Raoush and Khaled Asfar
The total molar flux equation for each component is
obtained by substitution of Equation (18) in the mass
balance Equations (i.e., Equation (14)) and utilizing
Equation (11). The total molar fluxes for component one,
two, and three are given by the Equations (19), (20) and
(21) below, respectively.
1
1
1
1
1
1
A = K ;B = K ;E = K ;H =
;G =
; Q=
D
D
D
D3
D1
D2
12
13
23
where gaseous components 1, 2 and 3 are chosen so
that M1 < M2 < M3 (Mi is the molecular weight of
component i ).
∇.N 1T = 0
• 2
• 2
• 2
• 2
• 2
• 2
•
•
α 1 X 2 + α 2 X 2 X 2 + α 3 X 22 X 2 + α 4 X 3 X 2 + α 5 X 2 X 3 X 2 + α 6 X 32 X 2 + α 7 X 2 X 3 +
•
•
•
•
•
•
•
• 2
• 2
•
•
•
α 8 X 2 X 2 X 3 + α 9 X 22 X 2 X 3 + α 10 X 3 X 2 X 3 + α 11 X 2 X 3 X 2 X 3 + α 12 X 32 X 2 X 3 +
• 2
• 2
• 2
• 2
••
α 13 X 3 + α 14 X 2 X 3 + α 15 X 22 X 3 + α 16 X 3 X 3 + α 17 X 2 X 3 X 2 + α 18 X 32 X 3 + α 19 X 2 +
••
••
••
••
••
(19)
••
α 20 X 2 X 2 + α 21 X 22 X 2 + α 22 X 23 X 2 + α 23 X 3 X 2 + α 24 X 2 X 3 X 3 + α 25 X 22 X 3 X 2 +
••
••
••
••
••
••
••
α 26 X 32 X 2 + α 27 X 2 X 32 X 2 + α 28 X 33 X 2 + α 29 X 3 + α 30 X 2 X 3 + α 31 X 22 X 3 + α 32 X 23 X 3 +
••
••
••
••
••
••
α 33 X 3 X 3 + α 34 X 2 X 3 X 3 + α 35 X 22 X 3 X 3 + α 36 X 32 X 3 + α 37 X 2 X 32 X 3 + α 38 X 33 X 3 = 0
∇.N 2T = 0
• 2
• 2
• 2
• 2
• 2
. 2
•
•
α 39 X 2 + α 40 X 2 X 2 + α 41 X X 2 + α 42 X 3 X 2 + α 43 X 2 X 3 X 2 + α 44 X X 2 + α 45 X 2 X 3 +
•
2
2
•
•
•
•
•
2
3
•
•
•
•
α 46 X 2 X 2 X 3 + α 47 X 22 X 2 X 3 + α 48 X 3 X 2 X 3 + α 49 X 2 X 3 X 2 X 3 + α 50 X 32 X 2 X 3 +
• 2
• 2
• 2
••
••
••
••
α 51 X 2 X 3 + α 52 X 22 X 3 + α 53 X 2 X 3 X 3 + α 54 X 2 + α 55 X 2 X 2 + α 56 X 22 X 2 + α 57 X 23 X 2 +
••
••
••
.••
••
••
α 58 X 3 X 2 + α 59 X 2 X 3 X 2 + α 60 X 22 X 3 X 2 + α 61 X 32 X 2 + α 62 X 2 X 32 X 2 + α 63 X 33 X 2 +
••
••
••
••
••
••
α 64 X 2 X 3 + α 65 X 22 X 3 + α 66 X 23 X 3 + α 67 X 2 X 3 X 3 + α 68 X 22 X 3 X 3 + α 69 X 2 X 32 X 3 = 0
- 250 -
(20)
Jordan Journal of Civil Engineering, Volume 1, No. 3, 2007
∇.N 3T = 0
• 2
• 2
• 2
•
•
•
•
•
•
α70 X 3 X 2 + α71 X 2 X 3 X 2 + α 72 X X 2 + α73 X 2 X 3 + α74 X 2 X 2 X 3 + α75 X 22 X 2 X 3 +
•
•
2
3
•
•
•
• 2
•
• 2
• 2
α76 X 3 X 2 X 3 + α77 X 2 X 3 X 2 X 3 + α 78 X 32 X 2 X 3 + α 79 X 3 + α 80 X 2 X 3 + α 81 X 22 X 3 +
• 2
• 2
• 2
••
••
••
••
••
••
(21)
α 82 X 3 X 3 + α 83 X 2 X 3 X 3 + α 84 X 32 X 3 + α 85 X 3 X 3 + α 86 X 2 X 3 X 2 + α 87 X 22 X 3 X 2 +
••
••
••
••
α 88 X 32 X 2 + α 89 X 2 X 32 X 2 + α 90 X 33 X 2 + α 91 X 3 + α 92 X 2 X 3 + α 93 X 22 X 3 + α 94 X 23 X 2 +
••
••
••
••
••
••
α 95 X 3 X 3 + α 96 X 2 X 3 X 3 + α 97 X 22 X 3 X 3 + α 98 X 32 X 3 + α 99 X 2 X 32 X 3 + α 100 X 33 X 3 = 0
where α is a coefficient function of DiK and Dij ,
values of α are given in the Appendix. Note from
Equation (11), for an n-component gaseous system, that
there are n-1 independent total molar flux equations.
Therefore, one equation out of Equations (19), (20) and
(21) is redundant. Equations (20) and (21) were chosen
to solve for the concentrations of the components. As
mentioned earlier, the fundamental idea behind
perturbation is to turn a difficult problem to a simple
one; this can be achieved by identifying terms in
Equations (20) and (21) that contribute insignificantly to
the equations. The less significant terms are then
i
i
k11 /( k1 +k11 )
1 ⎛ ( Z +ψ2 )(k1 + k11 ) ⎞
⎟⎟
X2 = ⎜⎜
ψ1
k11 ⎝
⎠
−
removed from the equations to reduce the non-linearity
of the equations. The relative importance of the terms in
Equations (20) and (21) was examined by calculating
the coefficients of the equations (i.e., α ) for different
ternary gaseous representing typical subsurface
contaminates and having widely different molecular
weights. Details of the procedure adapted in this study
to obtain first-order approximate solutions of the molar
fractions are presented in Appendix A. Results are
shown below:
The first-order approximate solution of X 2 as:
k10 α80
+ ( m7 Z2 + m8Z3 + m9Z4 )
k11 α95
i
(22)
The first-order approximate solution of X 3 can be found using the same procedure used to solve for X 2 . X 3
is given as:
⎛ ( Z +ψ 4 )( k23 + 1 ) ⎞
⎟
X 3 = ⎜⎜
⎟
ψ3
⎝
⎠
1 /( k23 +1 )
− k28 +
α 80
( m16 Z 2 + m17 Z 3 + m18 Z 4 )
α 95
(23)
For ternary gaseous system, Equation (11) is used to solve for X 1 :
n
∑X
i
= 1.0
(24)
i =1
where:
- 251 -
Perturbation Solution…
Wa'il Y. Abu-El-Sha'r, Riyadh Al-Raoush and Khaled Asfar
k1 = α39 /α95
k10 = α54 /α95
k11 = α55 /α95
k23 = α79 /α95
k28 = α91 /α95
m7 = {
(25)
(26)
(27)
(28)
(29)
k76
}
2
2k82
m8 = {
(30)
− k 76 k 81
k k
− 1 763 }
3
3 k 82
3 c 2 k 82
(31)
⎛ k2 k
kk k
k 2 k76 k k 2 k k k ⎞
⎟
m9 = ⎜⎜ 81 476 + 1 76 481 + 1 2 4 − 76 481 + 1 112 76
4 ⎟
3c2 k82 6c2 k82 12k82 12c2 k82
⎝ 3k82
⎠
⎞
⎛ k
m 16 = ⎜⎜ 974 ⎟⎟
⎝ 2 k 82 ⎠
(32)
(33)
⎛−k k
k k ⎞
m17 = ⎜⎜ 833 97 − 23 973 ⎟⎟
3c6 k84 ⎠
⎝ 3k84
(34)
⎛ k2 k
k k k
k2 k
k k2
k k
m 18 = ⎜⎜ 83 497 + 23 97 4 83 + 232 974 − 97 483 + 97 2 234
3 c 6 k 84
6 c 6 k 84
12 k 84
24 c 6 k 84
⎝ 3 k 84
k34
k44
k45
k46
k47
k59
k60
k61
k62
k63
⎞
⎟
⎟
⎠
= α58 /α80
= α59 /α80
= α61 /α80
= α62 /α80
= α63 /α80
= α92 /α80
= α96 /α80
= α98 /α80
= α99 /α80
= α100 /α80
(35)
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
(44)
(45)
⎛k k k
k k kk
k2 k k
k k k k2
k k3 k ⎞
k76 = ⎜⎜ 43 228 1 + 10 44 21 28 + 28 245 1 − 10 46 21 28 − 47 228 1 ⎟⎟
k 11 c2
c2
k 11 c6
c2
⎠
⎝ c2
⎛ k + k 11 ⎞
⎟⎟
k 81 = ⎜⎜ 1
⎝ c2
⎠
c
k 82 = 3 ( k1 + k11 )
c2
(46)
(47)
(48)
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Jordan Journal of Civil Engineering, Volume 1, No. 3, 2007
k 83 =
( k 23 + 1 )
c6
k84 =
(49)
c7
( k23 + 1 )
c6
(50)
⎛k k k
k k k k
k3 k k
k k k2 k k k2 k
k 97 = ⎜ 10 59 2 23 − 10 60 232 28 + 28 632 23 − 23 612 28 + 10 62 282 23
⎜ k11c6
k11c6
c6
c6
k11c6
⎝
⎞
⎟
⎟
⎠
(51)
α39 = (-a2c2d+abc2d-2a2cds+2abcds-a2ds2+abds2)
(52)
(53)
α54 = (-a2c2b – a2c2d-2a2bcs+2a2cds- a2bs2+a2 ds2)
(54)
α55 = (a2c2d -abc2d-2a2bck -2a2cdk+2a2bcs+4a2cds –2abcds-2a2bks –2a2dks+2a2bs2+3a2ds2-abds2)
(55)
α58 = (a2c2d – a2c2k+2a2bc – 2abc2s +4a2cds - 2ac2ds – 2a2cks+2a2bs2 –2abcs2+ 3a2ds2+ 2acds2- a2ks2)
α59 = (2a2cdk - ac2dk- a2ck2 -4a2cds+2abcds+2ac2ds –bc2ds+2a2bks –3abcks+ 4a2dks - 3acdks – a2k2s (56)
2a2bs2+2abcs2- 6a2ds2 +2abds2+ 4acds2- bcds2 +2a2ks2 - abks2)
(57)
α61 = (-2a2cds +2ac2ds-2a2cks -2ac2ks - a2cks2+2abcs2 –bc2s2 -3a2ds2 + 4acds2 - c2ds2+2a2ks2 - 2acks2)
α62 = (-2a2dks +3acdks - c2dks + a2k2s - ack2s+3a2ds2 – abds2 -4acds2 + bcds2 + c2ds2 - 2a2ks2 +abks2 + 2acks2 – bcks2)
(58)
(59)
α63 = (a2ds2 - 2acds2 + c2ds2 - a2ks2 + 2acks2 - c2ks2)
(60)
α79 = (-a2b2s + ab2cs -2a2bds +2abds - a2d2s + acd2s)
α80 = (abcdk +-acd2k - a2bk2 -2a2bds - 2ab2ds – 2abcds +b2cds +2a2d2s -2abd2s - 2acd2s + bcd2s +ab2ks +abdks)
(61)
(62)
α91 = (- a2b2c -2a2bcd -a2cd2 - a2b2s -2a2bds - a2d2s)
(63)
α92 = (2a2bcd -2ab2cd +2a2cd2 -2abcd2 - a2b2k -2a2bdk –a2d2k +a2b2s+ 4a2bds –2ab2ds +3a2d2s - 2abd2s)
α95 = (2a2bcd +2a2cd2 -2a2bck - 2a2cdk + a2b2s –ab2cs +4a2bds –2abcds + 3a2d2s - acd2s – 2a2bks - 2a2dks) (64)
α96 = (-2a2cd2 +2aabcd2 +2a2bdk + 2a2cdk –3abcdk +2a2d2k - acd2k– a2bk2 -a2dk2 - a2bds +2ab2ds +2abcds – b2cds
-6a2d2s +4abd2s +2acd2s – bcd2s + 2a2bks –ab2ks + 4a2dks –3abdks)
(65)
2 2
2
2 2
2
2 2
2
2
2
2 2
α98 = (-a cd +2a cdk -a ck -2a bds +2abcds -3a d s +2acd s +2a bks – 2abcks + 4a dks -2acdks - a k s) (66)
α99 = (-a2d2k +acd2k + a2dk2 +acd2k +3a2d2s -2abd2s –2acd2s + bcd2s - 4a2dks +3abdks+2acdks – bcdks +a2k2s abk2s)
(67)
(68)
α100 = (a2d2s -acd2s -2a2dks +2acdks +a2k2s + ack2s)
a =
b =
c =
d =
s =
k =
1
D
1k
1
D
2 k
1
D
(69)
3 k
1
D
12
1
D
13
1
D
23
- 253 -
Perturbation Solution…
Wa'il Y. Abu-El-Sha'r, Riyadh Al-Raoush and Khaled Asfar
incorporates the same transport mechanisms assumed in
this paper to solve for the DGM. The ternary gaseous
system used consisted of components that have widely
different molecular weights and exhibit no surface
diffusion. Physical properties of the components of the
system and boundary concentrations are given in Table
(1). Note that Dij and DiK are chosen to be approximately
of the same order of magnitude to provide equal
contribution from Knudsen and molecular mechanisms.
Figure (1) shows the concentrations of X A , X B and X C
plotted vs. Z ( Z = l L ) at P=100 atm obtained by
perturbation and numerical solutions. There is a good
agreement between the two solutions. However, in the
case of the numerical solution, concentration lines of X
and X B are curved in opposite direction. Remick and
Geankoplis (1970) reported that this behavior was
observed at high pressures, concentration lines tend to be
more linear as pressure decreases.
Here, a<b<c (i.e. M1 < M2 < M3 )
where, Mi is the molecular volume weight of i. Note
that Z in Equations (22) and (23) is a dimensionless
parameter: Z = l L , where l is a distance along the
diffusion path at which the concentrations to be
calculated (measured from the boundary) and L is the
total length of diffusion path.
Although the solution requires the evaluation of the
terms given in equations (22 – 69), it is a relatively
simple approach when compared to the reported
numerical solutions (Remick and Geankoplis, 1970). If a
spread sheet is prepared to evaluate these terms, the
solution can be obtained in a fast and effective manner
for different boundary conditions.
A
Solution Verification
To verify the perturbation solution of the DGM,
concentration profiles of components of a ternary gaseous
system were calculated using Equations (22) and (23) and
compared to concentrations obtained by an independent
numerical solution of a set of equations that describe the
transition region between Knudsen and molecular
diffusion. While the numerical solution obtained by
Remick and Geankoplis (1970) did not utilize the DGM
equations, it provides a base for verification because it
(N )
T
1
(N )
T
2
DGM
(N )
T
3
Ai =
=
DGM
DGM
=
=
− ∆12 ∆ 23 ∆ 31
X 1 ∆ 23 + X 2 ∆ 31 + X 3 ∆12
− ∆ 12 ∆ 23 ∆ 31
X 1 ∆ 23 + X 2 ∆ 31 + X 3 ∆ 12
− ∆12 ∆ 23 ∆ 31
X 1 ∆ 23 + X 2 ∆ 31 + X 3 ∆12
⎧ A1
⎪
⎪ ∆ 23
⎨
⎪
⎪
⎩
⎧ A2
⎪
⎪ ∆ 13
⎨
⎪
⎪
⎩
⎧ A3
⎪
⎪ ∆ 21
⎨
⎪
⎪
⎩
Application Examples
Equations (22), (23) and (11) can be used to calculate
mole fractions of ternary gaseous systems components. In
addition, the total molar fluxes can be calculated
explicitly for each component by incorporating the mole
fractions using the following equations (Jackson, 1977):
⎛ X2
X3
⎜
⎜ DK + DK
3
⎝ 2
X1
⎞
⎛ A2
A3
⎟ − X 1⎜
⎟
⎜ DK∆ + DK∆
3
12
⎠
⎝ 2 31
X
X
+ K2 + K3
K
D1
D2
D3
⎛ X1
⎛ A
X ⎞
A
⎜⎜ K + K3 ⎟⎟ − X 2 ⎜⎜ K 1 + K 3
D3 ⎠
D 3 ∆ 12
⎝ D1
⎝ D 1 ∆ 23
X3
X1
X2
+ K + K
D 1K
D2
D3
⎛ A
⎛ X1
X ⎞
A
⎜ K + K2 ⎟ − X 3 ⎜ K 1 + K 2
⎜D
⎟
⎜D ∆
D
D
2 ⎠
2 ∆ 31
⎝ 1
⎝ 1 23
X
X1
X
+ K2 + K3
D iK
D2
D3
P
∇X i
RT
⎞⎫
⎟⎪
⎟
⎠⎪
⎬
⎪
⎪
⎭
⎞⎫
⎟⎟ ⎪
⎠⎪
⎬
⎪
⎪
⎭
⎞⎫
⎟⎪
⎟
⎠⎪
⎬
⎪
⎪
⎭
(70)
(71)
(72)
(73)
- 254 -
Jordan Journal of Civil Engineering, Volume 1, No. 3, 2007
1
∆ ij
=
1
+
D ij
(74)
1
n
D
K
i
D
K
j
∑
Xi / D
K
i
i=1
(60), (61) and (11) are shown in Figure (3).
(3) The third example used is an experimental system
used by Abu-El-Sha’r (1993) to evaluate the relative
importance of different transport mechanisms in
gaseous systems. The system considered herein is an
open system where CH4 was injected at one side of a
soil sample placed in a diffusion cell and air (i.e., N2
and O2) was injected at the other side. System
properties and boundary conditions are given in
Table (4). The objective of this example is to
calculate molar fluxes of CH4 at different points for
different types of porous media. Three types of soil
were used; sea sand, Ottawa sand and Kaoliniate.
Figure (4) shows total molar fluxes for CH4 in the
different soil samples as calculated by perturbation
solution.
To test the validity of Equations (22-24), the
Equations are applied to scenarios including both
experimental and field applications as follows:
(1) The first example represents a field application
where the scenario and the data are borrowed from
Thorstenson and Pollock (1989a). This example
represents a typical subsurface contamination
problem in which a contaminant is generated at a
continuous rate at a specific depth below the ground
surface. One-dimensional transport is assumed (i.e.,
transport in the vertical direction). The system is
treated as a ternary system where Methane (CH4), the
major contaminant in this example, is considered as
one component and the major constituents of air (i.e.,
N2 and O2) are considered as two distinct
components. System properties and boundary
conditions are given in Table (2); note that Knudsen
diffusion is neglected in this example. Of particular
interest in such scenario is to calculate concentrations
(i.e., mole fraction) as a function of depth for the
different components in the system. Figure (2) shows
mole fractions of the components of the system as
obtained from the perturbation solution.
(2) The second example used is an experimental
diffusion system used by Karimi et al., (1987) to
investigate vapor phase diffusion of C6H6 in soil. The
experiment was designed to accommodate diffusion
cell contained soil sample extracted from cover of a
landfill used for disposal of industrial wastes. A
source of C6H6 was placed beneath the cell to mimic
the landfill by allowing diffusion through the soil
sample. At the top of the soil sample, an N2 source
was placed to sweep away C6H6. Properties and
boundary conditions of the system are shown in
Table (3). The objective of this example is to
calculate the concentration of C6H6 at different points
of the landfill cover. Mole fractions of the
components of the system as calculated by Equations
SUMMARY AND CONCLUSION
A perturbation method was used to solve the mass
balance equations for ternary gas systems when the
dusty gas model (DGM) is incorporated to calculate
mole fractions and molar fluxes of the components of
the system. Steady-state conditions, isobaric, isothermal
and non reactive system were assumed. The straightforward expansion method to the second-order
approximation was implemented in the solution. The
perturbation parameter which was introduced into the
equations, presented no physical meaning to the system.
The solution was expressed as a closed-form solution
incorporates a dimensionless length boundary
conditions, and parameters of transport mechanisms. The
solution was verified by a numerical solution and there
was a good agreement between the two solutions. The
perturbation solution was applied to different
experimental and field conditions to predict mass
fractions and molar fluxes of components of ternary
gaseous systems.
- 255 -
Perturbation Solution…
Wa'il Y. Abu-El-Sha'r, Riyadh Al-Raoush and Khaled Asfar
APPENDICES
Appendix A
Details of the Perturbation Solution of the DGM
Molar fluxes of species along a given direction (i.e., the z direction) in a multicomponent gaseous system are given
by Equation (15). For steady-state diffusive mass transport, it is commonly assumed that viscous flow and surface
diffusion are insignificant (Farmer et al., 1980; Karimi et al., 1987; Baeher and Bruell, 1990; Voudrious and Li, 1992).
Therefore, the first and third terms of the right hand side of Equation (15) may be neglected. Using Equations (15), (16)
and the relations: Pi=Xi P and Ci = Pi RT , the molar fluxes of a three-component gaseous system can be given as (i.e.,
Equation 17):
X3
X2
⎡ 1
⎡N1⎤
⎢DK + D + D
12
13
⎥
⎢
1
⎢
⎥
⎢
⎢
⎥
⎢
⎢
X2
⎢N ⎥
⎢
−
2
⎥ = −⎢
⎢
D 21
⎥
⎢
⎢
⎥
⎢
⎢
X3
⎥
⎢
⎢
−
N
⎢ 3⎥
⎢
D
31
⎥
⎢
⎢
⎦
⎣
⎣⎢
−
X1
D 12
X3
X1
1
+
+
D 21
D 23
D 2K
−
X3
D 32
−
X1
D 13
−
X2
D 23
X1
X2
1
+
+
D 31
D 32
D 3K
-1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦⎥
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣⎢
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦⎥
P •
X1
RT
P •
X2
RT
P •
X3
RT
(A-1)
Dots in the above equation represent the derivative with respect to z. For formulation convenience, let:
1
1
1
1
1
1
;G =
;Q =
A = K ;B = K ;E = K ;H =
D12
D13
D 23
D1
D2
D3
(A-2)
where gaseous components 1, 2 and 3 are chosen so that M1 < M2 < M3 (Mi is the molecular weight of component i ).
The total molar flux equation for each component is obtained by substitution of Equation (A-1) in the mass balance
n
Equations and utilizing
∑x
i
= 1.0 . The total molar fluxes for component one, two and three are given by the Equations
i =1
(A-3), (A-4) and (A-5), respectively.
∇ .N 1T = 0
• 2
• 2
• 2
• 2
• 2
• 2
•
•
α 1 X 2 + α 2 X 2 X 2 + α 3 X 22 X 2 + α 4 X 3 X 2 + α 5 X 2 X 3 X 2 + α 6 X 32 X 2 + α 7 X
•
•
•
α 8 X 2 X 2 X 3 + α 9 X 22 X
•
2
• 2
• 2
•
X 3 + α 10 X 3 X
•
2
•
X 3 + α 11 X 2 X 3 X
•
X 3 + α 12 X 32 X
• 2
• 2
• 2
•
2
X 3+
2
•
2
• 2
X 3+
••
α 13 X 3 + α 14 X 2 X 3 + α 15 X 22 X 3 + α 16 X 3 X 3 + α 17 X 2 X 3 X 2 + α 18 X 32 X 3 + α 19 X 2 +
••
••
••
••
••
••
α 20 X 2 X 2 + α 21 X 22 X 2 + α 22 X 23 X 2 + α 23 X 3 X 2 + α 24 X 2 X 3 X 3 + α 25 X 22 X 3 X 2 +
••
••
••
••
••
••
••
α 26 X 32 X 2 + α 27 X 2 X 32 X 2 + α 28 X 33 X 2 + α 29 X 3 + α 30 X 2 X 3 + α 31 X 22 X 3 + α 32 X 23 X 3 +
••
••
••
••
••
••
α 33 X 3 X 3 + α 34 X 2 X 3 X 3 + α 35 X 22 X 3 X 3 + α 36 X 32 X 3 + α 37 X 2 X 32 X 3 + α 38 X 33 X 3 = 0
- 256 -
(A-3)
Jordan Journal of Civil Engineering, Volume 1, No. 3, 2007
∇ .N 2T = 0
• 2
• 2
• 2
• 2
• 2
. 2
•
•
α 39 X 2 + α 40 X 2 X 2 + α 41 X 22 X 2 + α 42 X 3 X 2 + α 43 X 2 X 3 X 2 + α 44 X 32 X 2 + α 45 X 2 X 3 +
•
•
•
•
•
•
•
•
•
•
α 46 X 2 X 2 X 3 + α 47 X 22 X 2 X 3 + α 48 X 3 X 2 X 3 + α 49 X 2 X 3 X 2 X 3 + α 50 X 32 X 2 X 3 +
• 2
• 2
• 2
••
••
••
••
(A-4)
α 51 X 2 X 3 + α 52 X X 3 + α 53 X 2 X 3 X 3 + α 54 X 2 + α 55 X 2 X 2 + α 56 X X 2 + α 57 X X 2 +
2
2
••
••
••
2
2
.• •
3
2
••
••
α 58 X 3 X 2 + α 59 X 2 X 3 X 2 + α 60 X 22 X 3 X 2 + α 61 X 32 X 2 + α 62 X 2 X 32 X 2 + α 63 X 33 X 2 +
••
••
••
••
••
••
α 64 X 2 X 3 + α 65 X 22 X 3 + α 66 X 23 X 3 + α 67 X 2 X 3 X 3 + α 68 X 22 X 3 X 3 + α 69 X 2 X 32 X 3 = 0
∇ .N 3T = 0
• 2
• 2
• 2
•
•
•
•
•
•
α 70 X 3 X 2 + α 71 X 2 X 3 X 2 + α 72 X 32 X 2 + α 73 X 2 X 3 + α 74 X 2 X 2 X 3 + α 75 X 22 X 2 X 3 +
•
•
•
•
•
• 2
•
• 2
• 2
α 76 X 3 X 2 X 3 + α 77 X 2 X 3 X 2 X 3 + α 78 X 32 X 2 X 3 + α 79 X 3 + α 80 X 2 X 3 + α 81 X 22 X 3 +
• 2
• 2
••
••
• 2
••
••
(A-5)
••
α 82 X 3 X 3 + α 83 X 2 X 3 X 3 + α 84 X X 3 + α 85 X 3 X 3 + α 86 X 2 X 3 X 2 + α 87 X X 3 X 2 +
2
3
••
••
2
2
••
••
••
α 88 X 32 X 2 + α 89 X 2 X 32 X 2 + α 90 X 33 X 2 + α 91 X 3 + α 92 X 2 X 3 + α 93 X 22 X 3 + α 94 X 23 X 2 +
••
••
••
••
••
••
α 95 X 3 X 3 + α 96 X 2 X 3 X 3 + α 97 X 22 X 3 X 3 + α 98 X 32 X 3 + α 99 X 2 X 32 X 3 + α 100 X 33 X 3 = 0
where α is a coefficient function of DiK and Dij , values of α are given in the Appendix. Note from Equation (11),
for an n-component gaseous system, that there are n-1 independent total molar flux equations. Therefore, one equation
out of Equations (A-3), (A-4) and (A-5) is redundant. Equations (A-4) and (A-5) were chosen to solve for the
concentrations of the components. As mentioned earlier, the fundamental idea behind perturbation is to turn a difficult
problem to a simple one; this can be achieved by identifying terms in Equations (A-4) and (A-5) that contribute
insignificantly to the equations. The less significant terms are then removed from the equations to reduce the nonlinearity of the equations. The relative importance of the terms in Equations (A-4) and (A-5) were examined by
calculating the coefficients of the equations (i.e., α ) for different ternary gaseous representing typical subsurface
contaminates and having widely different molecular weights. Upon neglecting the insignificant terms, Equations (A-4)
and (A-5) are reduced to Equations (A-6) and (A-7), respectively.
i
i
i
∇ .N 2T = 0
• 2
• 2
•
•
•
•
•
•
•
•
α 39 X 2 + α 42 X 3 X 2 + α 45 X 2 X 3 + α 46 X 2 X 2 X 3 + α 48 X 3 X 2 X 3 + α 49 X 2 X 3 X 2 X 3 +
•
•
• 2
• 2
••
••
••
(A-6)
α 50 X 32 X 2 X 3 + α 51 X 2 X 3 + α 53 X 2 X 3 X 3 + α 54 X 2 + α 55 X 2 X 2 + α 58 X 3 X 2 +
••
••
••
••
.••
••
••
••
α 59 X 2 X 3 X 2 + α 60 X 22 X 3 X 2 + α 61 X 32 X 2 + α 62 X 2 X 32 X 2 + α 63 X 33 X 2 + α 64 X 2 X 3 +
α 67 X 2 X 3 X 3 + α 69 X 2 X 32 X 3 = 0
- 257 -
Perturbation Solution…
Wa'il Y. Abu-El-Sha'r, Riyadh Al-Raoush and Khaled Asfar
∇ .N 3T = 0
•
•
α 73 X
2
α 84 X
2
3
•
X 3 + α 76 X 3 X
• 2
X 3 + α 88 X
•
2
•
X 3 + α 78 X 32 X
••
X 3 + α 79 X 3 + α 80 X 2 X 3 + α 82 X 3 X 3 + α 83 X 2 X 3 X 3 +
••
••
••
••
X 2 + α 91 X 3 + α 92 X 2 X 3 + α 95 X 3 X 3 + α 96 X 2 X 3 X 3 + α 98 X
2
3
••
••
α 99 X 2 X 32 X 3 + α 100 X 33 X
• 2
• 2
• 2
• 2
•
2
2
3
(A-7)
••
X 3+
=0
3
The variables in Equations (A-6) and (A-7) are X 2 , X 3 and z ( X 2 and X 3 are dimensionless and z has a length unit).
To obtain dimensionless forms of Equations (A-6) and (A-7), the following variables are introduced:
∗
∗
∗
z
; X 2 = X2 ; X 3 = X3
L
The star in Equation (A-8) represents a dimensionless variable.
Z=
∗
(A-8)
∗
dX 2 d X 2 d Z
=
, Therefore:
∗
dz
d Z dz
∗
∗
d2X2
dX 2 1 d X 2
1 d2 X 2
=
;
and
=
∗ 2
dz
L d Z∗
dz 2
L2
dZ
•
(A-9)
••
Similarly, X 3 and X 3 can be expressed as:
∗
∗
dX 3 1 d X 3
d2X3
1 d2 X 3
=
; and
= 2
∗
2
∗ 2
dz
L dZ
dz
L
dZ
(A-10)
Dimensionless forms of Equations (A-6) and (A-7) are obtained by the following steps: (1) substituting Equations
(A-9) and (A-10) into Equations (A-6) and (A-7); (2) multiplying both sides of Equations (A-6) and (A-7) by L2; and
(3) dividing Equations (A-6) and (A-7) by α 95 , a relatively large coefficient chosen arbitrary to define a small
dimensionless perturbation parameter ∈ as:
∈=
α 80
α 95
(A-11)
Note that the perturbation parameter in this case has no physical meaning; it is artificially introduced into the equation
to reduce non-linearity. The dimensionless forms of Equations (A-6) and (A-7) are given by Equations (A-12) and (A-13),
respectively:
• 2
• 2
•
k 1 X 2 + k 34 ∈ X 3 X 2 + k 35 ∈ X
k 39 ∈ X
2
•
3
X
•
2
•
•
2
X 3 + k 36 ∈ X 2 X
• 2
• 2
•
2
•
X 3 + k 37 ∈ X 3 X
••
•
2
•
X 3 + k 38 ∈ X 2 X 3 X
••
••
X 3 + k 40 ∈ X 2 X 3 + k 41 ∈ X 2 X 3 X 3 + k 10 X 2 + K 11 X 2 X 2 + k 43 ∈ X 3 X 2 +
••
k 44 ∈ X 2 X 3 X 2 + k 45 ∈ X
••
2 ••
3
X 2 + k 46 ∈ X 2 X
k 49 ∈ X 2 X 3 X 3 + k 50 ∈ X 2 X
2 ••
3
X
3
2 ••
3 ••
3
3
X 2 + k 47 ∈ X
••
X 2 + k 48 ∈ X 2 X 3 +
=0
- 258 -
•
2
X 3+
(A-12)
Jordan Journal of Civil Engineering, Volume 1, No. 3, 2007
•
•
•
2 •
•
• 2
•
• 2
• 2
k 51 ∈ X 2 X 3 + k 52 ∈ X 3 X 2 X 3 + k 53 ∈ X 3 X 2 X 3 + k 23 X 3 + ∈ X 2 X 3 + k 54 ∈ X 3 X 3 +
• 2
2 • 2
••
2 ••
2 ••
••
••
••
k 55 ∈ X 2 X 3 X 3 + k 56 ∈ X 3 X 3 + k 57 ∈ X 3 X 2 + k 28 X 3 + k 59 ∈ X 2 X 3 + X 3 X 3 +
2 ••
(A-13)
3 ••
k60 ∈ X 2 X 3 X 3 + k61 ∈ X 3 X 3 + k62 ∈ X 2 X 3 X 3 + k63 ∈ X 3 X 3 = 0
where k i are dimensionless coefficents given in the Appendix. Note that X 2 , X 3 and Z are dimensionless and the
star is removed from the notation for convenience. The second-order approximation of X 2 and X 3 are given by
Equations (A-14) and (A-15), respectively:
X 2 ( Z ; ∈ ) = X 20 + ∈ X 21 + O ( ∈2 )
(A-14)
X 3 ( Z ; ∈ ) = X 30 + ∈ X 31 + O ( ∈2 )
(A-15)
where X i 0 and X i 1 are the zero-order and first-order approximations of X i , respectively.
Substitution of Equations (A-14) and (A-15) into equations (A-12) and (A-13) and expansion of all terms give the
following equations:
• 2
•
• 2
•
•
•
•
•
•
•
k1 X 20 + 2k1 ∈ X 21 X 20 + k34 ∈ X 30 X 20 + k35 ∈ X 20 X 30 + k36 ∈ X 20 X 20 X 30 + k37 ∈ X 30 X 20 X 30 +
•
2
•
•
• 2
•
• 2
••
k38 ∈ X 20 X 30 X 20 X 30 + k39 ∈ X 30 X 20 X 30 + k40 ∈ X 20 X 30 + k41 ∈ X 20 X 30 X 30 + k10 X 20 +
••
••
••
••
••
••
K10 ∈ X 21+ k11X 20 X 20 + k11 ∈ X 20 X 21+ k11 ∈ X 21 X 20 + k43 ∈ X 30 X 20 + k44 ∈ X 20 X 30 X 20 +
2 ••
2
••
3 ••
••
(A-16)
••
k45 ∈ X 30 X 20 + k46 ∈ X 20 X 20 X 30 + k47 ∈ X 30 X 20 + k48 ∈ X 20 X 30 + k49 ∈ X 20 X 30 X 30 +
2 ••
k50 ∈ X 20 X 30 X 30 + O(∈2 ) = 0
•
•
•
2
•
•
•
• 2
• 2
• 2
k51 ∈ X 20 X 30 + k52 ∈ X 30 X 20 X 30 + k53 ∈ X 30 X 20 X 30 + k23 X 30 + ∈ X 20 X 30 + k54 ∈ X 30 X 30 +
•
•
• 2
2
••
••
• 2
2 ••
••
••
2 ∈ k23 X 30 X 31 + k55 ∈ X 20 X 30 X 30 + k56 ∈ X 30 X 30 + k57 ∈ X 30 X 20 + k28 X 30 + k28 ∈ X 31 +
••
••
••
2 ••
(A-17)
k59 ∈ X 20 X 30 + X 30 X 30 + ∈ X 30 X 31 + ∈ X 31 X 30 + k60 ∈ X 20 X 30 X 30 + k61 ∈ X 30 X 30 +
2 ••
3 ••
k62 ∈ X 20 X 30 X 30 + k63 ∈ X 30 X 30 + O(∈2 ) = 0
Equating terms of like powers of ∈ in Equations (A-16) and (A-17) to zero gives two sets of equations: (1) when the
power of ∈ equals zero, Equations (A-18) and (A-19) are obtianed; and (2) when the power of ∈ equals one, Equations
(A-20) and (A-21) are obtianed.
• 2
••
••
k1 X 20 + k10 X 20 + k11 X 20 X 20 = 0
• 2
••
(A-18)
••
k23 X 30 + k28 X 30 + X 30 X 30 = 0
(A-19)
- 259 -
Perturbation Solution…
•
Wa'il Y. Abu-El-Sha'r, Riyadh Al-Raoush and Khaled Asfar
• 2
•
•
•
•
•
•
• 2
•
2k1 X 21 X 20 + k34 X 30 X 20 + k35 X 20 X 30 + k36 X 20 X 20 X 30 + k37 X 30 X 20 X 30 + k40 X 20 X 30 +
•
2
•
•
• 2
•
••
••
••
k38 X 20 X 30 X 20 X 30 + k39 X 30 X 20 X 30 + k41 X 20 X 30 X 30 + k10 X 20 + K10 X 21 + k11 X 20 X 21 +
••
••
2
••
••
••
• 2
• 2
2
3
••
(A-20)
k11 X 21 X 20 + k43 X 30 X 20 + k44 X 20 X 30 X 20 + k45 X 30 X 20 + k46 X 20 X 20 X 30 + k47 X 30 X 20 +
••
2
••
••
k48 X 20 X 30 + k49 X 20 X 30 X 30 + k50 X 20 X 30 X 30 = 0
•
•
•
•
2
• 2
2 ••
•
•
•
•
k51 X 20 X 30 + k52 X 30 X 20 X 30 + k53 X 30 X 20 X 30 + X 20 X 30 + k54 X 30 X 30 + 2k23 X 30 X 31 +
• 2
2
••
2 ••
••
••
••
••
k55 X 20 X 30 X 30 + k56 X 30 X 30 + k57 X 30 X 20 + k28 X 31 + k59 X 20 X 30 + X 30 X 31+ X 31 X 30 +
2 ••
(A-21)
3 ••
k60 X 20 X 30 X 30 + k61 X 30 X 30 + k62 X 20 X 30 X 30 + k63 X 30 X 30 = 0
The general solution of Equation (A-18) can be obtained as follows:
•
dX 20
Let w = X 20 =
dZ
By the chain rule:
(A-22)
d 2 X 20 dw
dw
dw dX 20
=
=
=w
2
dX 20
dZ dX 20 dZ
dZ
(A-23)
substitution of
d 2 X 20
dX 20
and
in Equation (A-18) gives:
dZ
dZ 2
dw
(k11 X 20 + k10 ) = −k1 w
dX 20
(A-24)
Integration of the above equation gives w as:
⎛ − k1 ⎞
⎜
⎟
⎜ k ⎟
11 ⎠
w = ψ 1 ( k 11 X 20 + k 10 )⎝
(A-25)
where ψ 1 is an arbitrary integration constant.
dX 20
for w gives the following equation:
Substituting
dZ
dX 20
= ψ 1 ( k11 X 20 + k10 )
dz
From Eqaution (42), X20 is obtianed as:
(A-26)
k 11 /( k 1 + k 11 )
⎛ ( Z + ψ 2 )( k1 + k11 ) ⎞
k
⎜⎜
⎟⎟
− 10
(A-27)
ψ
k 11
1
⎝
⎠
where ψ 2 is an arbitrary integration constant. In similar manner, using the same procedure used to obtian the general
solution of Equation (A-18) (i.e., Equations A-22- A-27), the general solution of Equation (A-19) can be obtained as:
X 20 =
X 30
1
k11
⎛ ( Z + ψ 4 )( k 23 + 1 ⎞
⎟
= ⎜⎜
⎟
ψ3
⎝
⎠
( k 23 + 1 ) −1
− k 28
(A-28)
- 260 -
Jordan Journal of Civil Engineering, Volume 1, No. 3, 2007
where ψ and ψ are arbitrary integration constants determined from boundary conditions. Substitution of Equations
(A-27), (A-28) and their deravitaves in Equation (A-20) gives the following equation:
3
••
X 21 +
4
•
[k 81 Z + k 82 ]−1 X 21 − 2k 1 k2 11 [k 81 Z + k 82 ]− 2 X 21 =
2k 1
ψ1
ψ1
k 64 [k 83 Z + k 84 ]
[k 81 Z + k 82 ]−( k + 2 k ) /( k + k ) + k 65 [k 81 Z + k 82 ]−( k + 2 k ) /( k + k ) +
−1
− k /( k + 1 )
− k /( k + k )
[k 83 Z + k 84 ]−k /( k +1 ) +
k 66 [k 81 Z + k 82 ] [k 83 Z + k 84 ]
+ k 67 [k 81 Z + k 82 ]
( 1− k ) /( k + 1 )
−1
− k /( k + k )
[k 83 Z + k 84 ]( 1−k ) /( k +1 ) +
k 68 [k 81 Z + k 82 ] [k 83 Z + k 84 ]
+ k 69 [k 81 Z + k 82 ]
−1
− 2 k /( k + 1 )
( 2 − k ) /( k + 1 )
k 70 [k 81 Z + k 82 ] [k 83 Z + k 84 ]
+ k 71 [k 83 Z + k 84 ]
+
( 1− 2 k ) /( k + 1 )
− k /( k + k )
− 2 k /( k + 1 )
[k 83 Z + k 84 ]
k 72 [k 81 Z + k 82 ]
+ k 73 [k 83 Z + k 84 ]
+
( 1− 2 k ) /( k + 1 )
1 /( k + 1 )
− k /( k + k )
−2
[k 83 Z + k 84 ]
k 74 [k 81 Z + k 82 ]
+ k 75 [k 81 Z + k 82 ] [k 83 Z + k 84 ]
+
−2
−2
2 /( k + 1 )
−2
3 /( k + 1 )
k 76 [k 81 Z + k 82 ] + k 77 [k 81 Z + k 82 ] [k 83 Z + k 84 ]
+ k78 [k 81 Z + k 82 ] [k 83 Z + k 84 ]
+
−( 2 k + 1 ) /( k + 1 )
− k /( k + k )
−( 2 k + 1 ) /( k + 1 )
[k 83 Z + k 84 ]
k 79 [k 83 Z + k 84 ]
+ k 80 [k 81 Z + k 82 ]
( k 23 + 1 )
−1
11
23
11
1
11
11
1
11
1
1
11
11
23
1
23
23
23
23
1
1
23
11
23
11
23
23
23
23
23
(A-29)
23
23
23
23
23
23
23
23
1
11
1
23
1
11
1
23
11
23
23
where k i are dimensionless coefficents given in the Appendix. In the homogenous part of Equation (A-29), define the
following functions:
p( Z ) =
q( Z ) =
2 k1
(A-30)
−2 k1 k 11
ψ ( k 81 Z + k 82 )2
(A-31)
ψ 1 ( k 81 Z + k 82 )
2
1
Since p( Z ) and q( Z ) are analytic and Z=0 is an ordinary point of Equation (A-29), the particular solution of Equation
(A-29) is given by the following form:
∞
X 21 ( Z ) = ∑ β j Z j
(A-32)
j =0
Substitution of Equation (A-32) in Equation (A-29) gives:
⎛ 2k 1 k11 ⎞
⎟β j Z j +
2
⎟
ψ
j =0 ⎝
1
⎠
∞
∑ ⎜⎜ −
∞
∑( k
2
82
) j( j − 1 )β j Z
⎛ 2k 1 k 82 ⎞
⎟⎟ jβ j Z j −1 +
ψ
j =1
1
⎠
∞
∑ ⎜⎜⎝
j −2
+
j =2
∞
∑ ( 2k
⎛ 2k1 k 81 ⎞
⎟⎟ jβ j Z j +
ψ
j =1
1
⎠
∞
∑ ⎜⎜⎝
k ) j( j − 1 )β j Z
81 82
j −1
∞
∑( k
2
81
) j( j − 1 )β j Z j +
j =2
(A-33)
= NHS
j =2
Note that NHS represents the non-homogenous side of Equation (A-29) (i.e., the right-hand side). Substitution of
η = j in the first, fourth and sixth summations, η = j − 2 in the second summations, and η = j − 1 in the third and fifth
summations of Equation (A-33) gives the following equation:
∞
∑( k
η =2
2
81
)η ( η − 1 )βη Z η +
∞
∑( k
η =0
2
82
)( η + 2 )( η + 1 )βη + 2 Z η +
∞
∑ ( 2k
η =1
k )( η + 1 )( η )βη +1 Z η +
81 82
⎛ ( 2k 1 k 82 ) ⎞
⎛ − 2 k 1 k11 ⎞
⎛ 2 k1 k 81 ⎞
η
⎟( η + 1 )βη + 1 Z η + ⎜
⎟
⎜⎜
⎟⎟ηβη Z η + ⎜
⎜ ψ 2 ⎟ βη Z = NHS
⎜ ψ
⎟
ψ
η =1 ⎝
η =0 ⎝
η =0 ⎝
1
1
⎠
1
⎠
⎠
∞
∑
∞
∑
∞
∑
- 261 -
(A-34)
Perturbation Solution…
Wa'il Y. Abu-El-Sha'r, Riyadh Al-Raoush and Khaled Asfar
Separating the terms corresponding to η = 0 , η = 1 , and combining the rest under one summation, gives:
⎛ 2k k
2
( 2k 82
)β 2 + ⎜⎜ 1 82
⎝ ψ1
⎛ − 2 k1 k11 ⎞
⎞
⎟
⎟⎟ β 1 + ⎜
⎜ ψ 2 ⎟β 0 +
⎠
1
⎝
⎠
⎛
⎛
4k k
2
⎜ ( 6 k 82
)β 3 + ⎜⎜ 4 k 81 k 82 + 1 82
⎜
ψ1
⎝
⎝
∑ (( k
∞
η =2
2
82
)( η + 2 )( η + 1 )βη + 2
⎛ 2k k
⎞
2k k ⎞ ⎞
⎟⎟ β 2 + ⎜ 1 81 − 1 2 11 ⎟ β 1 ⎟ Z +
⎜ ψ
ψ 1 ⎟⎠ ⎟⎠
1
⎠
⎝
⎛
⎞
2k k
+⎜⎜ ( 2k 81 k 82 )( η + 1 )( η ) + 1 82 ( η + 1 ) ⎟⎟ βη +1 +
ψ1
⎝
⎠
(A-35)
⎛ 2
⎞ ⎞ ⎞
⎛
⎞ ⎛
⎜ ( k 81 )( η − 1 )η + ⎜ 2 k1 k 81 ⎟η + ⎜ − 2k1 k11 ⎟ βη ⎟ Z η ⎟ = NHS
⎜ ψ ⎟ ⎜ ψ2 ⎟ ⎟ ⎟
⎜
1
⎝
⎠ ⎝
1
⎠ ⎠ ⎠
⎝
Equating the coefficients of like powers of Z on both sides of Equation (A-35) gives Equations (A-36 –A-38):
⎛ − 2 k1 k11 ⎞
⎛ 2k k ⎞
2
⎟ β 0 = k76
( 2k 82
)β 2 + ⎜⎜ 1 82 ⎟⎟ β 1 + ⎜⎜
(A-36)
2
⎟
⎝ ψ1 ⎠
⎝ ψ1 ⎠
⎛ 2k k
⎛
4k k ⎞
2k k ⎞
2
( 6 k 82
)β 3 + ⎜⎜ 4 k 81 k 82 + 1 82 ⎟⎟ β 2 + ⎜⎜ 1 81 − 1 2 11 ⎟⎟ β 1 = 0
ψ1 ⎠
ψ1 ⎠
⎝
⎝ ψ1
⎛
⎞
⎛ 2k k ⎞
( k 822 )( η + 2 )( η + 1 )βη + 2 + ⎜ ( 2k 81 k 82 )( η + 1 )( η ) + ⎜⎜ 1 82 ⎟⎟( η + 1 ) ⎟ βη + 1 +
⎜
⎟
⎝ ψ1 ⎠
⎝
⎠
⎛
⎞
⎛
⎞
⎜ ( k 2 )( µ − 1 )η + ⎜ 2k1 k 81 ⎟η + ⎛⎜ − 2k1 k11 ⎞⎟ ⎟ β = 0 ,
82
⎜
⎜ ψ 1 ⎟ ⎜⎝ ψ 12 ⎟⎠ ⎟ η
⎝
⎠
⎝
⎠
⎛ k
⎞ ⎛ k 1 k 11
⎟+⎜ 2 2
⎟ ⎜ψ k
⎠ ⎝ 1 82
⎛ − 2k k k
1 11 81
β 3 = ⎜⎜
2 3
3
ψ
1 k 82
⎝
⎞
⎛ − k1 ⎞
⎟β 0 + ⎜
⎟
⎜ ψ k ⎟β 1
⎟
⎝ 1 82 ⎠
⎠
⎛ 2k k
−ψ 1 k1 k 81 + k1 k11 ⎞
2k 2 k ⎞
2 k12
⎟β 1 +
+
− 13 113 ⎟⎟ β 0 + ⎜⎜ 81 21 +
2 2
2
⎟
3ψ 1 k 82 ⎠
3ψ 12 k 82
⎠
⎝ 3ψ 1 k 82 3ψ 1 k 82
⎛ − k76 k 81
k1 k76
⎜
⎜ 3k 3 − 3ψ k 3
82
1 82
⎝
(A-38)
η≥2
where β j , j = 1,...η is an arbitrary constant. Solving Equations (A-36 –A-38) for β 2 , β 3 and β 4 in terms of
gives:
β 2 = ⎜⎜ 762
⎝ 2k 82
(A-37)
⎞
⎟
⎟
⎠
- 262 -
β0
and
(A-39)
(A-40)
β1
Jordan Journal of Civil Engineering, Volume 1, No. 3, 2007
⎛ k2 k
kk k
kk k
k 2k
k k2
kk k
k k k ⎞ ⎛ 2k k k 2
β4 = ⎜⎜ 81 476 + 1 76 481 + 1 76 481 + 1 2 764 − 76 81
− 1 76 481 + 1 112 764 ⎟⎟ + ⎜⎜ 1 211 481 +
4
6ψ 1k82 6ψ 1 k82 12k82
6ψ 1k82 12ψ 1 k82 ⎠ ⎝ 3ψ 1 k82
3ψ 1k82
⎝ 3k82
2
2
⎛ − 2k81
2k12 k11k81 k12 k11k81
k13 k11
k1 k11k81
k12 k11k81
k12 k112 ⎞
k1 2k12 k81
⎟
⎜
β
+
+
−
−
+
+
4
4
4
4
4
4 ⎟ 0
⎜ 3ψ k 3 − 3ψ 2 k 3 +
3ψ 13 k82
3ψ 13k82
3ψ 14 k82
6ψ 12 k82
3ψ 13 k82
6ψ 14 k82
1 82
1 82
⎠
⎝
(A-41)
2
k1k81
k k k81
k 2k
k3
k 2k
k 2k
kk
k 2k
k 2k ⎞
− 1 11
− 1 2 813 − 31 3 + 1 2 813 − 1 3 113 + 1 2 813 + 1 2 813 − 1 3 113 ⎟⎟β1
2 3
2 3
3ψ 1 k82 3ψ 1 k82 3ψ 1 k82 3ψ 1 k82 6ψ 1 k82 6ψ 1 k82 6ψ 1 k82 3ψ 1 k82 6ψ 1 k82 ⎠
The general solution of Equation (A-29) can be expressed as:
X 21 = β0 ( 1 + m1Z 2 + +m2 Z 3 + m3 Z 4 + ...)+ β1( Z + m4 Z 2 + m5 Z 3 + m6 Z 4 + ...)+
( m7 Z 2 + +m8 Z 3 + m9 Z 4 + ...)
(A-42)
where mi are coefficients function of properties of system components (i.e., gases) given in the Appendix, β 0 and β 1
are arbitrary constants. Only the particular solution of Equation (A-29) is of interest, therefore, from Equation (A-42), X 21
can be given as:
X 21 = m7 Z 2 + m8 Z 3 + m9 Z 4
(A-43)
Substitution of Equations (A-27) and (A-43) in Equation (A-14) gives the first-order approximate solution of X 2 as:
1 ⎛ ( Z + ψ 2 )( k1 + k11 ) ⎞
⎜
⎟⎟
k11 ⎜⎝
ψ1
⎠
X2 =
k 11 /( k 1 + k 11 )
−
k10 α 80
+
( m7 Z 2 + m8 Z 3 + m9 Z 4 )
k11 α 95
(A-44)
The first-order approximate solution of X 3 can be found using the same procedure used to solve for X 2 . X 3 is
given as:
1 /( k 23 + 1 )
⎛ ( Z + ψ 4 )( k 23 + 1 ) ⎞
α
⎟
X 3 = ⎜⎜
− k 28 + 80 ( m16 Z 2 + m17 Z 3 + m18 Z 4 )
⎟
ψ3
α 95
⎝
⎠
For ternary gaseous system, Equation (A-46) is used to solve for X 1 :
n
∑X
i
= 1.0
(A-45)
(A-46)
i =1
Note that Z in Equations (A-44) and (A-45) is a dimensionless parameter: Z = l L , where l is a distance alnog the
diffusion path at which the concentrations to be calculated (measured from the boundary), and L is the total length of
diffusion path.
α39= (-A2E2H+ABE2H-2A2EHG+2ABEHG-A2HG2+ABHG2).
α54= (-A2BE2-A2E2H-2A2BEG-2A2EHG-A2BG2-A2HG2).
α55= (A2E2H-ABE2H-2A2BEQ-2A2EHQ+2A2BEG+4A2EHG-2ABEHG-2A2BQG2A2HQG+2A2BG2+3A2HG2-ABHG2).
α58= (A2E2H-A2E2Q+2A2BE-2ABE2G+4A2EHG-2AE2HG-2A2EQG+2A2BG22ABEG2+3A2HG2-2AEHG2-A2QG2).
α59= (2A2EHQ-AE2HQ-A2EQ2-4A2EHG+2ABEHG+2AE2HG-BE2HG+2A2BQG+2A2EQG3ABEQG+4A2HQG-3AEHQG-A2Q2G-2A2BG2+2ABEG2-A2HG2+2ABHG2+4AEHG2-BEHG2+2A2QG2-ABQG2).
α61= (-2A2EHG+2AE2HG+2A2EQG-2AE2QG-A2BG2+2ABEG2-BE2G2-3A2HG2+4AEHG2-
- 263 -
Perturbation Solution…
Wa'il Y. Abu-El-Sha'r, Riyadh Al-Raoush and Khaled Asfar
E2HG2+2A2QG2-2AEQG2).
α62= (-2A2HQG+3AEHQG-E2HQG+A2Q2G-AEQ2G+3A2HG2-ABHG24AEHG2+BEHG2+E2HG2-2A2QG2+ABQG2+2AEQG2-BEQG2).
α63= (A2HG2-2AEHG2+E2HG2-A2QG2+2AEQG2-E2QG2).
α79= (-A2B2G+AB2EG-2A2BHG+2ABEHG-A2H2G+AEH2G).
α80= (ABEHQ+AEH2Q-A2BQ2-A2HQ2+2A2BHG-2AB2HG-2ABEHG+B2EHG+2A2H2G2ABH2G-2AEH2G+BEH2G+AB2QG+ABHQG).
α91= (-A2B2E-2A2BEH-A2EH2-A2B2G-2A2BHG-A2H2G).
α92= (2A2BEH-2AB2EH+2A2EH2-2ABEH2-A2B2Q-2A2BHQ-A2H2Q+A2B2G+4A2BHG2AB2HG+3A2H2G-2ABH2G).
α95= (2A2BEH+2A2EH2-2A2BEQ-2A2EHQ+A2B2G-AB2EG+4A2BHG-2ABEHG+3A2H2GAEH2G-2A2BQG-2A2HQG).
α96= (2A2EH2+2ABEH2+2A2BHQ+2A2EHQ-3ABEHQ+2A2H2Q-AEH2Q-A2BQ2-A2HQ2A2BHG+2AB2HG+2ABEHG-B2EHG-6A2H2G+4ABH2G+2AEH2G-BEH2G+2A2BQG-AB2QG+4A2HQG-3ABHQG).
α98= (-A2EH2+2A2EHQ-A2EQ2-2A2BHG+2ABEHG-3A2H2G+2AEH2G+2A2BQG2ABEQG+4A2HQG-2AEHQG-A2Q2G).
α99= (-A2H2Q+AEH2Q+A2HQ2+AEH2Q+3A2H2G-2ABH2G-2AEH2G+BEH2G4A2HQG+3ABHQG+2AEHQG-BEHQG+A2Q2G-ABQ2G).
α100= (A2H2G-AEH2G-2A2HQG+2AEHQG+A2Q2G+AEQ2G).
k1 = α39 / α95
k10 = α54 / α95
k11 = α55 / α95
k23 = α79 / α95
k28 = α91 / α95
k43 = α58 / α80
k44 = α59 / α80
k45 = α61 / α80
k46 = α62 / α80
k47 = α63 / α80
k59 = α92 / α80
k60 = α96 / α80
k61 = α98 / α80
k62 = α99 / α80
k63 = α100 / α80
2
2
3
⎛ k43k28k1 k10k44k1k28 k28
k45k1 k10k46k1k28
k47 k28
k1 ⎞
⎜
⎟
k76 = ⎜ −
+
+
−
−
2
2
2
2
2
⎟
k
k
ψ
ψ
ψ
ψ
ψ
1
11 1
1
11 3
1
⎝
⎠.
⎛k +k ⎞
k 81 = ⎜⎜ 1 11 ⎟⎟
⎝ ψ1 ⎠ .
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Jordan Journal of Civil Engineering, Volume 1, No. 3, 2007
k 82 =
k 83 =
ψ2
(k1 + k11 )
ψ1
.
( k 23 + 1)
ψ3
.
ψ
k 84 = 4 (k 23 + 1)
ψ3
.
⎛k k k
k k k k
k k3 k
k2 k k
k k k2 k ⎞
k 97 = ⎜⎜ 10 59 2 23 − 10 60 282 23 + 63 282 23 − 28 232 61 + 10 62 282 23 ⎟⎟
k11ψ 3
ψ3
ψ3
k11ψ 3
⎝ k11ψ 3
⎠.
⎛ k ⎞
m7 = ⎜⎜ 762 ⎟⎟
⎝ 2k82 ⎠ .
⎛−k k
k k
m8 = ⎜⎜ 763 81 − 1 763
3ψ 1 k 82
⎝ 3k 82
⎞
⎟
⎟
⎠.
⎛k2 k
k k k
k 2k
k k2 k k k
m9 = ⎜⎜ 81 476 + 1 76 481 + 1 2 764 − 76 481 + 1 112 764
3ψ 1 k 82
6ψ 1 k 82 12k 82 12ψ 1 k 82
⎝ 3k 82
⎞
⎟
⎟
⎠.
⎛ k ⎞
m16 = ⎜⎜ 972 ⎟⎟
⎝ 2k 84 ⎠ .
⎛−k k
k k ⎞
m17 = ⎜⎜ 833 97 − 23 973 ⎟⎟
3ψ 3 k 84 ⎠
⎝ 3k 84
.
2
⎛k k
k k k
k2 k
k k2
k k
m18 = ⎜⎜ 83 497 + 23 97 483 + 232 974 − 97 483 + 97 2 234
3ψ 3 k 84
6ψ 3 k 84 12k 84 24ψ 1 k 84
⎝ 3k 84
⎞
⎟
⎟
⎠.
- 265 -
Perturbation Solution…
Wa'il Y. Abu-El-Sha'r, Riyadh Al-Raoush and Khaled Asfar
Table 1: Properties and boundary conditions of the system used to
verify the perturbation Solution.
Boundary Concentrations
Component
i
Mi
g/mole
DiK
cm2/sec
Dij
cm2/sec
A = He
4.003
0.837207
DAB = 1.126
B = Ne
20.183
0.372868
DAC = 0.729
XB0 = 0.0
XBL=0.5
C = Ar
39.944
2.65033
DBC = 0.322
XC0 = 0.5
XCL=0.5
z=0
z=L
XA0 = 0.5
XAL=0.0
Table 2: Properties and boundary conditions of the system given in Example 1.
Component
Mi
Dije †
Boundary Concentrations
Dij
2
2
i
g/mole
cm /sec
cm /sec
z=0
z = L (10m)
A = CH4
B = N2
C = O2
16.043
28.013
31.999
DAB = 0.2137
DAC = 0.2263
DBC = 0.2083
DAB = 0.02137
DAC = 0.02263
DBC = 0.02083
XA0 = 1.0
XB0 = 0.0
XC0 = 0.0
XAL=0.0
XBL=0.78
XCL=0.22
† Dije is calculated as Dije = 0.1Dij .
Table 3: Properties and boundary conditions of the system given in Example 2.
Boundary Concentrations
Component
Mi
DiK †
Dije §
Dij ‡
z=L
i
g/mole
cm2/sec
cm2/sec
cm2/sec
z=0
(L=2.54 cm)
A = N2
28.013
0.0125
DAB = 0.284
0.0426
XA0 = 0.702
XAL=1.0
B = O2
31.999
0.0117
DAC = 0.107
0.0161
XB0 = 0.198
XBL=0.0
C = C6H6
78.114
0.0075
DBC = 0.100
0.0150
XC0 = 0.10
XCL=0.0
is calculated using Eq. 3 ( r =4X10-7 cm for clayey soil (Abriola et al., 1990), T=20oC).
†D
‡ Data from Abu-El-Sha’r, 1993.
§ Dije is calculated using Eqn. (8) and (9), assuming T p = 0.45 .
K
i
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Jordan Journal of Civil Engineering, Volume 1, No. 3, 2007
Table 4: Properties and boundary conditions of the system given in Example 3.
Qp
Mi
DiK †
Dij
gas
Qm
Soil
Porosity
(cm)
g/mole
cm2/sec
cm2/sec
DAB=
A=CH4
16.043
424.7
0.2137
Sea
DAC=
0.46
0.22
0.0109
B=N2
28.013
321.4
Sand
0.2263
DBC=
C=O2
31.999
300.74
0.2083
DAB=
A=CH4
16.043
1180.7
0.2137
Ottawa
DAC=
0.36
0.17
0.0303
B=N2
28.013
893.5
sand
0.2263
DBC=
C=O2
31.999
836.01
0.2083
DAB=
A=CH4
16.043
31.17
0.2137
DAC=
Kaolinite
0.85
0.41
0.0008
B=N2
28.013
23.59
0.2263
DBC=
C=O2
31.999
22.07
0.2083
† DiK is calculated using Eqn. (2) (T= 20 oC).
‡ Dije is calculated using Eqn. (8).
Perturbation solution
Numerical solution
0.5
XC
0.4
XA
Xi 0.3
XB
0.2
0.1
0
0
0.2
0.4
0.6
Z=(l/L)
0.8
1
Figure 1: Comparison between perturbation and numerical solutions for a
ternary gaseous system.
- 267 -
Dije ‡
cm2/sec
0.0470
0.0497
0.0458
0.0363
0.0384
0.0354
0.0876
0.0927
0.0850
Perturbation Solution…
Wa'il Y. Abu-El-Sha'r, Riyadh Al-Raoush and Khaled Asfar
1
CH4
N2
O2
0.8
0.6
Xi
0.4
0.2
0
0
2
4
6
8
10
z (m)
Figure 2: Mole fractions of the components of the system given in Example 1 as calculated
by perturbation solution.
1
N2
O2
C6H6
0.8
0.6
Xi
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
z (cm)
Figure 3: Mole fractions of the components of the system given in Example 2 as calculated
by perturbation solution.
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Jordan Journal of Civil Engineering, Volume 1, No. 3, 2007
0.14
Kaolinite
Sea sand
Ottawa sand
0.1
T
-2
-1
N (g m sec )
0.12
0.08
0.06
0.04
1
1.5
2
2.5
3
3.5
4
4.5
5
z (cm)
Figure 4: Total molar fluxes of CH4 in different porous media systems given in Example 3
as calculated by perturbation solution.
Notation
A
coefficient (L2 t-1)
B
coefficient (L2 t-1)
c
total molar concentration (mole L-3)
ci
molar concentration of component i (mole L-3)
Dij
free binary diffusivity of gases i and j (L2 t-1)
Dije
effective binary diffusion coefficient of gases i and j (L2 t-1)
D iK
Knudsen diffusivity (Knudsen diffusion coefficient) of gas i (L2 t-1)
DiK ( r )
Knudsen diffusivity of gas i in a pore of radius r (L2 t-1)
Dis ( r )
effective surface diffusivity of component i in a pore of radius r (L2 t-1)
E
coefficient (L2 t-1)
G
coefficient (L2 t-1)
H
coefficient (L2 t-1)
i
index
j
index
k
intrinsic permeability (L2)
ki
coefficient
- 269 -
Perturbation Solution…
Wa'il Y. Abu-El-Sha'r, Riyadh Al-Raoush and Khaled Asfar
L
length (L)
Mi
molecular weight of component i (M mole-1)
mi
coefficient
n
number of gas components
n'
molar density of gas and particles (mole L-3)
N iD
total molar diffusive flux of component i (mole L-2 t-1)
N iK
Knudsen molar flux of component i (mole L-2 t-1)
N iT
total molar flux of component i (mole L-2 t-1)
Nv
molar viscous flux (mole L-2 t-1)
[N z ]
column vector with elements N iz ,..., N nz
P
pressure (M L-1 t-2)
Q
coefficient (L2 t-1)
Qm
obstruction factor (diffusivity)
QP
obstruction factor for Knudsen diffusivity (the effective Knudsen radius) (L)
R
ideal gas constant (M L2 t-2 T-1 mole-1)
Rvi
reaction rate of component i per unit volume of porous media (M3 t-1 L-3)
r
pore radius (L)
r
average pore radius (L)
T
temperature (T)
t
time (t)
TP
total porosity of the porous media
Xi
mole fraction of component i
∗
X
Z
dimensionless mole fraction
dimensionless length, l L
∗
Z
z
dimensionless length
length (L)
DGM
dusty gas model
VOCs
volatile organic compounds
NHS
nonhomogenous side
αi
coefficient
α ij
generalized thermal diffusivity
βi
coefficient
η
index
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Jordan Journal of Civil Engineering, Volume 1, No. 3, 2007
∈
perturbation parameter
ε
void fraction
µ
dynamic viscosity (M L-1 t-1)
ψi
integration constant
ΣV
sum of atomic diffusion volumes
∞
infinity
∇
gradient operator
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Wa'il Y. Abu-El-Sha'r, Riyadh Al-Raoush and Khaled Asfar
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