3.2. Inert gas stripping method
3.2.1. Equations for IDACs computation
3.2.1.1. Equations proposed by Leroi et al.
(1977)
3.2.1.1(a) General equations proposed by Leroi et al.
(1977)
Leroi et al. (1977) derived basic equations to compute
infinite dilution activity coefficients from the dilutor technique. The
starting point of their formulation is the assumption that the two phases in
the cell are in thermodynamic equilibrium. The equilibrium equations for each
of the components are:
(3-19)
(3-20)
(3-21)
where the subscripts refer to the solute (1), the carrier gas
(2) and the solvent (3), is the mole
fraction in the liquid phase, is the mole fraction in the vapour
phase, is the pressure, is
the fugacity, is the fugacity coefficient, is the reference
fugacity for a liquid at pure state
and zero pressure, is the activity coefficient, is the Henry`s
law constant and is the
Poynting correction given by:
(3-22)
In the above equation, is the molar volume of the component at
temperature. The
L
reference state is the pure component at zero pressure for both
solvent and solute. At low
v
pressure, (conditions at which all the experiments in this
study were carried out), the vapour phase corrections can be derived from
second virial coefficients. Fugacity coefficients, which can be determined at
low pressures from second virial coefficients, are used to account for the
vapour phase imperfections as follows:
(3-23)
(3-24)
is the system temperature, is the virial coefficient related to
bimolecular interactions between and molecules, is the mixed second virial
coefficient at temperature . R is
the gas constant. The equation below can be used to determine
reference fugacity values, that are required in equations (3-19) and (3-20).
(3-25)
where is the reference molar volume of liquid component, is the
vapour pressure of
component and is the fugacity coefficient in the vapour phase at
saturation. If the solute is
considered infinitely dilute in the solvent and
the carrier gas is insoluble or of negligible
solubility in the liquid phase, then the activity coefficient
of the solute, can be equated to its infinite dilution activity coefficient in
the solvent (3), and both the activity coefficient of the solvent, and its mole
fraction, in the liquid phase to 1. Neglecting vapour phase imperfections,
equilibrium equations (3-19) and (3-20) can be written as follows:
(3-27)
Amounts of solute and solvent removed from the equilibrium cell
by the carrier gas flow can be calculated with the aid of these equations:
(3-28)
(3-29)
where and are respectively the number of moles for the solute
and the solvent in the
equilibrium cell at time , and give the change in the amount of
the two components
with time. is the total volumetric rate of the gas phase leaving
the still, converted to pressure
and temperature . A combination of equations (3-26) through
(3-29) leads to:
(3-30)
(3-31)
From overall mass balance calculations around the equilibrium
cell, the total volumetric rate
of the gas phase leaving the still can be related to, the pure
carrier gas flow rate measured at system pressure and system temperature.
(3-32)
Combining equations (3-30) through (3-32) gives:
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Chapter 3: Theoretical considerations
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(3-33)
(3-34)
(3-35)
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1 ?
At infinite dilution, it can be deduced that
N P P
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The differential equations (3-30) and (3-31) describe the
variations of the number of moles in
the cell with time for both the solute
and the solvent. Substituting in these two equations by
equation (3-33) and replacing by (3-35) result in:
(3-36)
(3-37)
where is the saturated vapour pressure of component . The last
two equations have been
derived using the following three assumptions:
· the vapour phase is ideal;
· The carrier gas solubility in the liquid phase is
negligible;
· The solute is infinitely diluted in the solvent.
In quest of simplification, Leroi et al. (1977) added more
assumptions depending on the volatility of the solvent.
3.2.1.1(b) Equation proposed by Leroi et al. (1977) for
non-volatile solvents.
A non-volatile solvent is one with negligible vapour pressure
equal or less than 1 mmHg (George 2008). Leroi et al. (1977) added the
following assumptions with reference to equations (3-36) and (3-37):
· is constant due to the non-volatile nature of the
solvent;
. The ratio is negligible with respect to 1 since the vapour
pressure of the
solvent is close to zero;
. The term is ignored due to the infinite dilution
assumption.
Thus, equation (3-37) does not exist any longer whereas
equation (3-36) can be written as
(3-39)
where is the initial number of moles for the solute in the cell.
During infinite dilution activity
coefficient measurements, it is advisable to maintain the gas
sampling valve at constant temperature and to ensure that the GC detector
linearity is satisfied. If these requirements are met, the amount of the solute
injected into the column given by the corresponding peak area from the GC, will
be proportional to its partial pressure over the solution.
(3-40)
is the proportionality constant depending on the GC used. From
equations (3-26), (3-35), (3-39) and (3-40), an equation showing an exponential
decrease of the solute peak area with
? = ln
13 DP t
sat ( )
A
time can be obtained.
(3-41)
is the initial solute peak area. Finally, the infinite
dilution activity coefficient of the solute (1) in the solvent (3) is
determined by means of this equation derived from expression (3-41):
(3-42) where N is the number of solvent moles in the dilutor
cell and expresses the initial solute peak area.
3.2.1.1(c) Equation proposed by Leroi et al. (1977) for
volatile solvents
When a volatile solvent is used, its amount in the cell varies
with time. Therefore, the two
equations (3-36) and (3-37) are worked out simultaneously to
give the following solution. n
(3-43)
3.2.1.2. Equation proposed by Duhem and Vidal (1978) for
non volatile solvents
Leroi et al. (1977) derived equation (3-43) on the basis of some
assumptions such as neglecting the term appearing in equations (3-36) and
(3-37). Duhem and Vidal established that
for large infinite dilution activity coefficients, this
particular assumption is no longer valid. Apart from dropping this assumption,
they expressed the mole fraction of the solute in the liquid phase by
(3-44)
This allowed them to come up with the following equation:
(3-45)
?
?
? 1 ?
Assuming a constant inert gas flow rate
N ? 1 ? , after incorporating a vapour
space correction into
? ? ?
? RT N ?
equation (3-45), Duhem and Vidal derived the following equation
which is more accurate for systems with large activity coefficients at infinite
dilution:
? ? (3-46)
? 1
? V 1 V 1
G
?
where is given by:
(3-47)
In the above equation and are the number of moles of the solute
in the still at a certain time and the initial number of moles for the solute
respectively. is the volume of the vapour
phase evaluated from equation
(3-48) (Bao et al. 1993a, b).
(3-48)
3.2.1.3. Equation proposed by Bao and Han (1995) for a
volatile solvent
Following the example of Duhem and Vidal (1978), Bao and Han
(1995) used equation (3-44)
/ ? 1 Psat / P?
to express the solute mole fraction in the liquid phase.
Neglecting the correction term suggested by Bao and Han, they solved the
differential equations (3-36) and (3-37) to obtain the following final result,
applicable to a volatile solvent:
?
(3-49)
3.2.1.4. Equation proposed by Hovorka and Dohnal
(1997)
According to Leroi et al. (1977), under the assumption that
the solvent is non-volatile, the partial pressure of the solute is negligible
as compared to the total pressure in the cell, the vapour phase is ideal and
neglecting the effect of the vapour space, the limiting activity coefficient
can be determined as follows:
(3-50)
For a volatile solvent, a presaturation cell is used as
explained in section 2.3.2. It is Hovorka and
Dohnal`s statement that after
presaturation the flow rate of the stripping gas is increased by a
factor where is the saturation vapour pressure of the solvent.
Thus, for the
double cell technique, the activity coefficient at infinite
dilution is computed as
(3-51)
These last two equations provide the first order approximation
of the infinite dilution activity
coefficient, denoted as , with a
sufficient degree of accuracy in most cases. However, for
improved accuracy, the second order approximation can be
used.
(3-52)
where is given by equation (3-50) or (3-51). Correction factors
are defined as follows:
· , correction factor associated with the change of the
inert stripping gas flow rate due
to the saturation in the cell is given for the single cell
technique (Equation 3-53) and the double cell technique (Equation 3-53)
respectively by:
(3-53)
and
(3-54) where is the mean amount of the solute in the cell
during the measurement and is given by:
(3-55)
is the initial amount of the solute in the cell, and are solute
peak areas at the
? ?
end and the beginning of the experiment, respectively.
k ? is always less than unity and
2 RT
rises with increasing volatility of the solvent and the
solute.
· is the correction factor associated with solvent removal
due to its volatility. When a presaturation cell is not used, this correction
term is written as:
(3-56)
where is the total stripping time, denotes the initial amount
of the solvent in the cell
and D is the stripping gas flow rate. If the stripping gas is
saturated, will become equal
to unity; it is always less than unity and
rises with decreasing cell volume and increasing
3 ? 3 ? 3
solvent volatility.
· is related to the amount of the solute in the vapour
space above the solution in the
cell. It is calculated by
(3-57)
where is the vapour space volume. is greater than unity and
rises with the solute
volatility and with the increasing ratio of the vapour space
volume to the amount of the solvent in the equilibrium cell.
· is the vapour phase nonideality correction factor given
by:
(3-58)
where subscripts =1` and =2` refer to the solute and the
stripping gas, respectively. is
the pure solute molar volume, is the second virial coefficient.
Depending on the
system and experimental conditions, can be either greater or
smaller than unity.
3.2.1.5. Equation proposed by Krummen et al.
(2000)
Krummen et al. (2000) proposed a formulation which takes into
account the saturation fugacity of the components under investigation and the
increase of the stripping gas flow rate caused by its saturation with the
solvent. Assuming that the liquid phase is in equilibrium with the gas phase in
the cell, these equations can be written:
(3-59)
(3-60)
Further assumptions are:
· The solute is highly dilute in the solvent. This implies
that:
(3-61)
(3-62)
(3-63)
· Measurements are carried out at low pressures or pressure
differences . Thus,
(3-64)
· The stripping gas is of negligible solubility in the
liquid phase. Therefore, (3-65) And for the solvent, it is assumed that:
(3-66)
The above simplifications allow writing equations (3-59) and
(3-60) as follows:
(3-67)
(3-68)
The flow rate of the stripping gas stream entering the dilutor
cell is made up of the flow rate
of the stripping gas stream entering the saturation cell,and the
flow rate of the solvent stream, .
(3-69)
The amount of solvent leaving the presaturator is provided by its
saturation vapour pressure and mixes with the pure stripping gas entering the
dilutor cell.
(3-70)
Combining equations (3-68) through (3-70) leads to:
(3-71)
y
According to Krummen et al. (2000), the solvent content in the
stripping gas stream cannot be dt RT
neglected when is greater than 5 mbar. , the flow rate of the
stripping gas stream
leaving the dilutor cell, is given by:
(3-72) where is the flow rate of the solute removed from the
measurement cell, which under the assumption of an ideal gas, can be calculated
as follows:
(3-73)
decreases with time. When the last two equations are combined, it
follows that:
(3-74) In order to determine activity coefficients at infinite
dilution by the inert gas stripping technique, the variation of the amount of
solute in the dilutor cell is measured as a function of time. This variation
can be expressed by the following equation:
(3-75)
Since the stripping gas is saturated with the solvent,
(3-76)
Taking into account equation (3-75), equation (3-74) can be
written as:
(3-77)
Combining equations (3-77) and (3-67) results in:
(3-78)
When equation (3-77) is used in equation (3-75) and combined with
equation (3-67), the result is:
(3-79)
For relatively volatile solutes, high saturation vapour
pressures of the solvent or large infinite dilution activity coefficient
values, Krummen et al. (2000) suggest that only the solute content in the
liquid phase should be taken into account when determining the solute molar
fraction.
n 1
x
(3-80)
Assuming ideal gas behaviour, the content of the solute in the
gas phase can be calculated as follows:
(3-81)
When equations (3-81) and (3-67) are combined and the resulting
expression is inserted into equation (3-80), the result is:
(3-82)
If equation (3-82) is used in equation (3-79), the variation of
the amount of solute with time can
? ?
1 1 1
? ? sat sat
P V ? ? ? ? st sat
P n RT
1
be expressed by:
?
NRT ?
(3-83)
In the above equation, is the corrective term used to take
into
account the variation of the solute gas stream during
measurements. If the change in the solute flow rate is neglected with respect
to the inert gas stream, the corrective term is equated to unity. In this
particular case, the integration of equation (3-83) yields:
(3-84)
If the linearity of the detector is assured and condensation
effects in the sample loop and other k n
tubes are avoided, the solute peak area A, is given
by:
(3-85)
k is the proportionality constant. When equations (3-82), (3-67)
and (3-85) are combined, the result is:
(3-86)
All the quantities on the right--hand side of this equation do
not vary during the measurements
? ?
except the number of moles for the solute,n,
Using from equation (3-86) in equation (3-84)
leads to:
(3-87)
Solving the equation for gives:
(3-88)
where is the slope of the graph of the natural logarithm of the
solute peak area versus time.
(3-89)
Using equation (3-71), expression (3-88) can be written as:
(3-90)
where is the vapour phase volume and
T P is the stripping gas flow rate as it enters the
FM
presaturator at the system temperature .
Equation (3-90) is also applicable to solvent mixtures with the
aid of these formulae:
(3-91)
(3-92)
where and are respectively saturation vapour pressures and
amounts of individual
solvents involved in the solution. Equation (3-90) shows that
activity coefficients at infinite dilution can be determined from the
experimentally measured slope, the saturation vapour pressures of the solute
and the solvent, the saturation fugacity coefficient of the solute, the system
pressure and temperature, the inert gas flow rate and the vapour phase volume.
The inert gas flow rate needs to be converted to the cell conditions using the
following equations, depending on the type of flowmeter used:
(Electronic flowmeter) (3-93)
(Soap bubble flowmeter) (3-94)
where is the inert gas flow rate at the flowmeter, is the
pressure at the flowmeter,
is the saturation vapour pressure for water, is the pressure in
the measurement cell. Equation (3-94) was also useful for determining the
corrected flow of the carrier gas when using GLC.
3.2.2. Mass Transfer considerations in the equilibrium
cell.
Richon et al. (1980) developed a useful model to calculate mass
transfer of the solute from the
solution to the vapour phase in the still. A
condition of validity of their method is that
thermodynamic equilibrium
should be achieved between the saturated gas leaving the still and
the liquid. Assuming that there is a very quick diffusion of
solute into the bubble, the modified Raoult`s law describes the existing
vapour-liquid equilibrium in the still, bubbles are perfectly spherical and the
carrier gas is not soluble in the solvent, they derived equations (3-95) and
(3-96) for the attainment of equilibrium in the liquid phase and in the vapour
phase respectively.
(3-95)
(3-96) where is the ratio of mass transfer in the cell to mass
transfer to reach equilibrium, taking into consideration only the liquid phase
resistance is the same as , taking into consideration gas phase diffusion only.
As the system approaches equilibrium, as well as approach
unity. is the path length of the bubbles in the equilibrium
still. is the density in g .cm-3,
is the temperature in Kelvin, is the diffusion coefficient of
solute, , in solvent,, in the
gas phase, is the bubble radius, is the solvent molar mass in
g/mol., is the solute
vapour pressure at temperature in atm and is the
limiting bubble speed given by
(3-97)
, kinematic viscosity in cP and is the diameter of bubbles. Many
authors have designed
dilutor cells on the basis of mass transfer calculation proposed
by Richon et al. (1980).
Of particular importance is equation (3-95) which allows the
determination of the optimal cell height (Li et al. 1993). It has been used for
this end in this work. Infinite dilution activity coefficients values were
computed using equation (3-90). Saturation fugacity coefficients were
determined from second virial coefficients (See appendix D)
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