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Multicomponent
adsorption equilibria
Ali Ahmadpour
Chemical Eng. Dept.
Ferdowsi University of Mashhad
2
Contents
Introduction
Langmuir Multicomponent Theory
The Basic Thermodynamic Theory
Ideal Adsorption Solution Theory (IAS)
Fast IAS Theory
Real Adsorption Solution Theory (RAST)
Potential Theory
Conclusions
3
Introduction
In most cases, adsorption systems usually involve
more than one component, therefore adsorption
equilibria involving competition between
molecules of different type is needed for
understanding the system as well as the design
purposes.
4
Langmuir Multicomponent
Theory
The extended Langmuir isotherm equation is the
simplest theory for describing multicomponent
equilibria.
This equation has assumed that each species maintains its own surface
area (i.e. the area covered by one species is not affected by the others).
5
Cont.
The selectivity of species "i" in relation to the
species "j" is defined as:
It is reminded that the extended Langmuir isotherm is derived assuming
the surface is ideal and the adsorption is localized, and one molecule can
adsorb onto one adsorption site.
6
Empirical Approaches Based on
the Langmuir Equation
When there are lateral interactions between adsorbed species, which are
generally different from the self interactions among molecules of the
same type, the following empirical equation is recommended:
The parameter ηi allows for interaction among molecules of the same
type, and ni allows for other effects of heterogeneity.
This equation is the generalization of the Langmuir-Freundlich (Sips)
isotherm.
7
The Basic Thermodynamic Theory
Recognizing the deficiency of the extended
Langmuir equation, despite its sound theoretical
footing on basic thermodynamics and kinetics
theories, other approaches have been proposed.
ideal adsorbed solution theory of Myers and Prausnitz,
real adsorption solution theory,
vacancy solution theory,
potential theory.
8
Ideal Adsorption Solution Theory
(IAS)
n°j : number of mole of pure component j per unit mass of the adsorbent;
Фj : surface potential of the pure component j;
xi, yi: mole fraction in the adsorbed phase and gas phase, respectively.
9
Cont.
If the pure component isotherm n°i(Pi) is known, the surface potential
of the pure component i can be obtained from the integral.
When IAS theory apply to the usual case of adsorption equilibria, we
should specify the total pressure and the mole fractions in the gas
phase and then determine the properties of the adsorbed phase which is
in equilibrium with the gas phase.
Then, when the gas phase is specified the total number of unknowns is:
10
Cont.
Once the total adsorbed amount is determined, the adsorbed phase
concentration of the component i is:
In theory of IAS, adsorption isotherm equation for pure components
can take any form which fits the data best.
Two isotherms commonly used are Toth and Unilan equations.
Toth equation is preferred from the computational point of view
because it usually gives faster convergence than the Unilan equation.
11
Cont.
The success of the calculations of IAS depends on how
well the single component data are fitted, especially in the
low pressure region as well as at high pressure region
where the pure hypothetical pressure lies.
An error in these regions, particularly in the low pressure
region, can cause a large error in the multicomponent
calculations.
12
Practical Considerations of The
IAS Equations
The IAS theory is recast in the form convenient for the
purpose of computation.
For a system containing N species, the IAS equations are:
P°i() is the hypothetical pressure of the pure component that gives the
same spreading pressure on the surface.
13
Cont.
The spreading pressure is the negative of the surface potential.
For a given total pressure (P) and the mole fraction in the gas
phase (yi), these equations provide a total of 2N+1 equations.
With this set of 2N+1 equations, there are 2N+1 unknowns.
N values of mole fractions in the adsorbed phase (xi),
one value of the spreading pressure π, and
N values of the hypothetical pressure of the pure component, P°j,
which gives the same spreading pressure as that of the mixture.
Solving the equations numerically will give a set of solution for
the adsorbed phase mole fractions and a solution for the
spreading pressure.
14
Cont.
For general case, the IAS equations must be
solved numerically and this can be done with
standard numerical tools, such as the Newton-
Raphson method for the solution of algebraic
equations and the quadrature method for the
evaluation of integral.
15
IAS Theory
The IAS theory provides a convenient means to
calculate the multicomponent equilibrium.
It has a number of attractive features:
The theory does not require any mixture data,
It is independent of the actual model of physical adsorption,
The form of adsorption isotherm equation for pure component
data is arbitrary (hence any equations that best fit the data can
be used),
Different solutes can have different forms of the isotherm
equation.
16
Cont.
Despite the versatility of the IAS theory, one disadvantage
is the evaluation of the reduced spreading pressure.
With the exception of few isotherm equations, the integral
equation for the spreading pressure is generally evaluated
numerically.
Even when the spreading pressure can be analytically
evaluated, the inverse problem of obtaining the pure
component "hypothetical“ pressure in terms of the reduced
spreading pressure is not always available analytically.
17
Fast IAS Theory
An isotherm, introduced by O'Brien and Myers (1984), is
obtained as a truncation to two terms of a series expansion of the
adsorption integral equation in terms of the central moments of
the adsorption energy distribution.
Using this equation which allows the computation of reduced
spreading pressure analytically is called the Fast IAS procedure.
Any isotherm that has explicit expression for the spreading
pressure can be used in the Fast IAS formalism.
18
Cont.
The difference between IAS and Fast IAS is that:
The Fast IAS involves the solution of N variables of pure
component pressure, bjPj0, while the solution for the spreading
pressure is sought in the IAS theory.
Once the spreading pressure is known in the IAS theory,
the hypothetical pure component pressures can be obtained
as the inverse of the integral equation.
Thus, solving directly for the hypothetical pressures of
pure component makes the computation using the Fast IAS
faster than the IAS theory.
19
Cont.
Although the Fast IAS is attractive to obtain the set of
solution for the hypothetical pressures, the applicability of
this theory is restricted to only a few equations which yield
analytical expressions for the reduced spreading pressure.
For other equations, the IAS theory has to be used instead.
20
Real Adsorption Solution
Theory (RAST)
The procedure used in IAS theory is simple and the
method of calculation is also straight forward.
Unfortunately, the method only works well when the
adsorption systems do not behave too far from ideality.
However for some nonideal cases (azeotropic), the IAS
theory is inadequate to describe the systems.
In such systems, the activity coefficient is used to account
for the nonideality of the adsorbed phase.
This procedure called as the real adsorption solution theory
(RAST).
21
Cont.
The activity coefficients can be either calculated from known
theoretical correlations or from binary equilibrium data.
They are a function of the composition of the adsorbed phase and
the spreading pressure. To calculate them, we need the binary
experimental data:
(a) the total pressure of the gas phase and its compositions
(b) the compositions of the adsorbed phase
(c) the spreading pressure of the mixture
Knowing these activity coefficients, they are then used in some
theoretical models, such as the Wilson equation to derive the binary
interaction parameters of that model.
22
Binary adsorption curves
G a s p h a se m o le fra c tio n , Y1
0 .0 0 .2 0 .4 0 .6 0 .8 1 .0
Am
ou
nt
ad
so
rbe
d (
mm
ole
/g)
0 .0
0 .5
1 .0
1 .5
2 .0
2 .5
3 .0N u x it c a rb o n , 2 9 3 K
C2H
6
C H4
IH F L
IA S T
M P S D
E D
S ip s
G a s p h a se m o le fra c tio n , Y1
0 .0 0 .2 0 .4 0 .6 0 .8 1 .0
Am
ou
nt
ad
so
rbe
d (
mm
ole
/g)
0 .0
0 .5
1 .0
1 .5
2 .0
2 .5
3 .0
A ja x ca rb o n , 2 7 3 K
C O2
C H4
IH F L
M P S D
E D
IA S T
S ip s
23
Binary adsorption curves
(X-Y diagrams)
M e th a n e in th e a d s o rb e d p h a s e , X1
0 .0 0 .2 0 .4 0 .6 0 .8 1 .0
Me
tha
ne
in
th
e g
as
ph
as
e,
Y1
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
C H4-C O
2
(a )
M e th a n e in th e a d s o rb e d p h a se , X1
0 .0 0 .2 0 .4 0 .6 0 .8 1 .0
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
(b )
C H4-C
2H
6
M e th a n e in th e a d s o rb e d p h a s e , X1
0 .0 0 .2 0 .4 0 .6 0 .8 1 .0
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
C H4-C
3H
8
(c )
24
Potential Theory
The extension of the potential theory was studied by Doong
and Yang (1988) to multicomponent systems.
The approach is simple in concept, and it results in analytical
solution for the multicomponent adsorption isotherm.
The basic assumption of the model is that:
there is no lateral interaction between molecules of different types,
pure component isotherm data are described by the DA equation.
25
Conclusions
Theories for adsorption equilibria in multicomponent systems
are not as advanced as those for single component systems.
The slow progress in this area is due to a number of reasons:
(i) lack of extensive experimental data for multicomponent
systems,
(ii) solid surface is too complex to model adequately.
However, some good progress has been steadily achieved in
this area.
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