Absorption of hydrogen sul&de into aqueous solutions of ferric
nitrilotriacetic acid: local auto-catalytic e*ects
J. F. Demmink, A. Mehra ∗ , A. A. C. M. BeenackersX
Department of Chemical Engineering, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
Abstract
The rates of reactive absorption of H2 S into aqueous solutions of ferric nitrilotriacetic acid, at T = 303 K and pH = 4:5, studied in a
9at-interface stirred cell, appear to be auto-catalyzed by freshly precipitated sulfur particles. These auto-catalytic e*ects, which are more
prominent at higher ferric chelate concentration, seem to involve particle-to-interface adhesion phenomena. A model based on Higbie’s
penetration theory, which incorporates particle-to-interface adhesion, as well as a growing particle coverage during a liquid element’s
contact time at the interface, is used to analyze the experimental data. This model gives a reasonable description of the local auto-catalytic
e*ects on the gas absorption rate.
Keywords: Absorption; Auto-catalytic; Particle; Kinetic
1. Introduction
Gas desulfurization with iron chelates is a commercial
process based on liquid redox chemistry, where ironcomplexes are alternately reduced and oxidized according to
(Oostwouder & Hodge, 1995; Nagl, 1997)
H2 S(g) H2 S(aq);
H2 S + 2Fe3+ Ln− → 2Fe2+ Ln− + 2H+ + S ↓;
(1)
O2 (g) O2 (aq);
O2 + 4Fe2+ Ln− + 2H2 O → 4Fe3+ Ln− + 4OH− :
(2)
Here n denotes the charge of an organic ligand L, which
usually is a polyaminocarboxylic acid, such as, ethylenediaminetetraacetic acid, (EDTA), hydroxyethylethylenediaminetriacetic acid (HEDTA), diethylenetriaminepentaacidic acid (DTPA) or nitrilotriacetic acid (NTA). Since
no iron chelate is consumed in the overall reaction, it may be
considered as a pseudo-catalyst. A comprehensive review
of this technology has been given by McManus and Martell
(1997).
Although desulfurization processes based on reactions (1)
and (2) have found widespread applications in industry, actual kinetic data on reaction (1) have only recently become
available, and the information is often limited to single phase
reaction systems (Lonergan, Lieberman, & Lewis, 1995).
These studies are of relevance for determining the reaction mechanisms but some important features of gas–liquid
mass transfer are not accounted for. Wubs and Beenackers
(1994) studied the absorption of H2 S into aqueous solutions
of ferric chelates of EDTA and HEDTA in a stirred cell reactor, and, although no direct evidence could be presented,
they suggested that in situ produced sulfur particles may
enhance the H2 S absorption rate. Earlier, gas absorption
enhanced by fresh sulfur particles had been observed by
Mehra and Sharma (1988), who studied the reactive absorption of H2 S in aqueous iodide=iodine (I − =I2 ) solutions in
a stirred cell reactor. These investigators found the absorption rates of H2 S to increase by a factor of 4–8, compared
to the case of physical absorption. They also discussed the
auto-catalytic enhancement by in situ produced sulfur particles, including the in9uence of electrolyte, Na2 SO4 , which
decreased the enhancement, and the e*ect of a surface
active agent, disodium-2-2 -dinaphtyl-methane-6,6 -disulfonate (TAMOL), which caused the enhancement by sulfur
to vanish. On the other hand, the use of iso-propanol led to
an increase in the rate-enhancing e*ect of the sulfur precipitates. It was postulated that whereas TAMOL deactivates
1724
the sulfur surface by possibly adsorbing on it, iso-propanol
changes the sulfur precipitation habit and a reduction in the
particle agglomeration tendency. These authors also developed a semi-empirical model which was based on adsorption
of H2 S on &ne sulfur particles, in combination with agglomeration of the particles to form coarse, inactive, lumps. This
approach was criticized by Wubs, Beenackers, and Krishna
(1991) who showed that the model proposed by Mehra
and Sharma (1988) was inadequate because it required
m-values (H2 S-distribution coeJcient, partitioning between
solid sulfur and liquid phase) that are unrealistically high
even if multilayer adsorption is assumed. They suggested
that an enhanced di*usion mechanism might be involved
and=or particle concentrations may be considerably higher
near the gas–liquid interface than in the bulk liquid. The
latter phenomenon is likely to occur in aqueous solutions
when hydrophobic particles are involved (Wimmers & Fortuin, 1988; Vinke, Bierman, Hamersma, & Fortuin, 1991;
Vinke, Hamersma, & Fortuin, 1993). This conclusion was
also drawn by Beenackers and van Swaaij (1993) in their
review paper.
In an earlier study we have shown that the absorption
rate of a non-reactive gas (acetylene) is enhanced by a
factor of 2–4 in the presence of small amounts of fresh
sulfur precipitates, even though the overall capacity of the
liquid phase appears to be unchanged (Demmink, Mehra,
& Beenackers, 1998). This study, which also includes similar experiments with carbon particles, supports the suggestion of Wubs et al. (1991) that interfacial adhesion of sulfur
particles may play an important role in enhancing the absorption of H2 S into reactive aqueous ferric chelate solutions.
A model, based on Danckwerts’ surface renewal theory
(Danckwerts, 1970), accounting for the interfacial aJnity of
particles was also presented by Demmink et al. (1998). This
model gave a reasonable semi-quanitative description of the
experimental observations thus providing some insights into
the near-interface phenomena involved in the rate enhancing
mechanisms.
Not accounted for in the above models, are the possible e*ects of near-interface generation (and accumulation)
of adhering particles as a result of chemical reaction (1)
during the contact time of a penetration element with
the gas–liquid interface. As demonstrated by Saraph and
Mehra (1994), such transient e*ects may be signi&cant.
These authors showed that even non-adhering particles,
being generated by chemical reaction close to the gas–
liquid interface, could be present within the surface element in concentrations much greater than in the liquid
bulk.
In this paper, the results of gas absorption experiments that demonstrate the e*ect of locally produced
sulfur on the rate of absorption accompanied by chemical reaction (1), are presented. Also, a model is developed which accounts for a growing coverage at the
interface by adhering particles being produced by reaction (1).
Fig. 1. Stirred cell reactor used for gas absorption experiments.
2. Experimental
The H2 S absorption experiments were carried out in
a glass, stirred reactor with a 9at gas–liquid interface as
shown in Fig. 1. A centrally located shaft equipped with
two six-bladed stirrers was used to mix the gas as well as
the liquid. Four symmetrically placed baMes were installed
to increase the e*ectiveness of mixing and to reduce the
formation of a vortex. Valve 1 was connected to an Olivetti
M240 computer enabling automatic reactor operation. A
pressure indicator (Druck) and a thermocouple were connected to the computer for automatic data collection. The
computer was equipped with a Burr Brown PCI 20089 W-1
interface.
The absorbing solutions were prepared by dissolving NTA
(Acros Chimica, 97%), FeCl3 ·6H2 O (GPR, ¿99%) and
NaOH (Acros Chimica) in water (reverse osmosis quality).
Background salt NaCl (0:2 kmol=m3
) was added to keep the
ionic strength of the solution (1=2
Ci zi2 = 1:0 kmol=m3 ,
where Ci is the concentration of ionic species i, and zi
its charge) practically constant during the experiments.
Sorbitol (0:5 mol=mol ferric ion) was added to prevent
precipitation of ferric hydroxide. The concentration CFe(II)
was determined by titration with Ce(SO4 )2 solution, using
ferroine as an indicator.
First the reactor was thoroughly evacuated to degass the
liquid completely. Then valve 1 was opened to feed pure nitrous oxide (N2 O or A ), until a pre-set pressure of 90 kPa
was attained. The liquid phase was then stirred until equilibrium was reached. The solubility of N2 O (HeA ) and the volumetric mass transfer coeJcient (kL; A a) were determined
from these experiments with N2 O which typically lasted for
approximately 3 h. The value for HeA was used to estimate
the solubility of H2 S in the ferric chelate solutions, HeA using the calculation procedure reported earlier (Demmink &
1725
Table 1
Reactor dimensions and reaction conditions
Total reactor volume, m3
Gas-phase volume, m3
Liquid impeller, m
Gas impeller, m
a (9at surface),
m2 =m3 (liquid)
Ns ; 1=s
T; K
pH
PA , Pa
CFe ; kmol=m3
CNTA =CFe , dimensionless
a For
1:77 × 10−3
0:77 × 10−3
Six-bladed turbine, dLs = 4 × 10−2
−2
Six-bladed turbine, dG
s = 6 × 10
8.37a
20
303
4.5
2:5 × 103
0:050–0:400
1
VL = 1:0 × 103 m3 .
The values for =0 were used for calculating the di*usivity
of H2 S (DA ) in the reaction liquid. The measured values for
and =0 , and the resulting values for DA and kL are given
in Table 2.
Particle sizes were determined with a Coulter Counter.
It was necessary to dilute the slurry-samples, immediately upon withdrawal, by an electrolyte solution
(1 mass % NaCl), containing 0:05 kg=m3 surfactant N -cetyl
N; N; N -trimethyl ammonium bromide (CTMAB). The procedure was found to e*ectively suppress particle agglomeration after sampling.
3. Results and observations
Beenackers, 1997; Weisenberger & Schumpe, 1996). Also,
from the measured kL; A a, the value of the liquid side mass
transfer coeJcient (kL; A ) was estimated which was then
used to calculate the kL -value for H2 S using
DA
kL; A = kL; A
:
(3)
DA In the following text kL will be used for denoting kL; A .
After evacuating the system, valve 1 was opened to feed
pure H2 S (AGA, 99%) until a certain pre-set pressure was
reached. Valve 1 was then closed and the gas allowed to absorb; the pressure consequently dropped until a minimum,
also pre-set, value was reached. Then valve 1 was opened
and the pressure increased. In this manner a sawtooth-like
time–pressure pro&le was recorded over a period of time,
from which the value for dPA =dte could be obtained. Since
the experiments were performed batch-wise with respect to
the liquid, the ferric chelate concentration CFe(III) decreased
during an experiment. Samples were drawn from the liquid
to measure pH and CFe(II) . Table 1 summarizes the conditions under which the experiments were carried out.
An Ubbelohde viscometer was used to determine viscosities of the reaction liquids used in this study, relative to the
viscosity of pure water, (=0 ). This method requires
the liquid density () which was determined by measuring
the apparent weight of a completely submerged glass body.
Consider the experiment using a solution with an initial
ferric NTA concentration of CFe(III); 0 = 400 mol=m3 . The
dPA =dte -values were recorded as a function of decreasing
CFe(III) . The value of the enhancement factor, EAexpt , de&ned
by
EAexpt =
JAexpt
i ;
kL CA;
L
(4)
was determined from the measured rate, estimated from
dPA
VG
expt
JA =
−
;
(5)
RT VL
dte
i
i
3
where CA;
L was calculated from HeA , as CA; L = 0:2 mol=m .
The result is shown in Fig. 2, line 5. Repeating this experiment with an identical, clean, solution, now using
CFe(III); 0 = 300 mol=m3 , results in a lower initial EAexpt -value,
which again increases with decreasing CFe(III) , within the
experiment. The procedure was repeated with clean solutions with initial concentrations CFe(III); 0 = 200; 100
and 50 mol=m3 , respectively. All the results are shown in
Fig. 2.
The initial EAexpt -values decrease with decreasing CFe(III); 0 .
The phenomenon of increasing EAexpt with increasing conversion becomes less pronounced as CFe(III); 0 is lowered and
&nally tends to vanish. The decrease in the initial EAexpt -value
is consistent with the expressions for the enhancement of
the gas absorption rates by chemical reaction, given in the
Table 2
Measured values of densities (), and viscosities relative to pure water (=0 ), for reaction liquids used in this study, as well as the liquid phase
di*usivities (DA ) and liquid side mass transfer coeJcients (kL ) for H2 S, as estimated from =0 , at T = 303 K; pH = 4:5
CFe(III) kmol=m3
0.05
0.1
0.2
0.3
0.4
a ;
1.122
1.47
1.57
3.5
1.139
1.60
1.48
3.5
1.148
1.70
1.41
4.7
1.160
1.95
1.28
3.5
1.191
2.42
1.09
2.3
kg=m3
dimensionless
DA ; m2 =s
d 105 k m=s
L
b ,
0
c 109
a Determined
from the apparent weight of a completely submerged glass body.
with Ubbelohde viscosimeter.
c D0 = 2:09 × 10−9 m 2 =s, taken from Haimour and Sandall (1984), T = 303 K. Di*usivities of H S were calculated according to (D =D0 ) = (0 =)0:6 .
2
A A
A
d Determined from the value for N O (k
2
L; A ), using Eq. (3).
b Determined
1726
expt
Fig. 2. EA
as a function of CFe(III) ; values of kL and DA
given in Table 2, T = 303 K; pH = 4:5. (1) CFe(III); 0 = 50 mol=m3 ;
(2) CFe(III); 0 = 100 mol=m3 ; (3) CFe(III); 0 = 200 mol=m3 ; (4)
CFe(III); 0 = 300 mol=m3 ; (5) CFe(III); 0 = 400 mol=m3 ; (6) “Fresh” solutions
(CFe(III) = CFe(III); 0 ); CS = 0; (7) line obtained by interpolation through
the initial EA -values (CFe(III) = CFe(III); 0 ). For readability, dashed lines are
included to connect the observations within one absorption experiment.
literature (Danckwerts, 1970), namely,
EA = (1 + Ha2 );
where, Ha is the Hatta number de&ned as
√
DA k2 CFe(III)
D A k1
=
Ha =
kL
kL
(6)
(7)
and k2 is the second-order reaction rate constant.
A similar trend may be expected, thus, within one given
experiment, that as the conversion of CFe(III) increases leading to a fall in its concentration, the EAexpt -value should also
go down. However, this is not the case, and within a given
experiment, the EAexpt -values increase with conversion, as can
be seen from Fig. 2. This presumably is on account of the
sulfur particles that form during the course of a given experiment and which are obviously absent at the commencement
of a fresh experiment of any initial CFe(III) .
Gas absorption experiments with a non-reactive gas such
as acetylene or a mildly reactive one like nitrous oxide (that
does not induce any absorption rate enhancement due to
the reaction) before and after the reactive H2 S absorption,
proved that the solubility (expressed by HeA ) of these gases
in the aqueous iron NTA solutions was not a*ected by the
sulfur particles produced by reaction (1). The rate of absorption of these non-reactive gases, however, increased by
a factor of almost 2 in the presence of sulfur particles. These
e*ects have been described in detail in our earlier publication (Demmink et al., 1998). After addition of small amounts
of surfactants (proprietary LO-CAT surfactant ARI 600 and
CTMAB, approximately 10−6 kmol=m3 ), the absorption rate
Fig. 3. E*ect of pH on gas absorption enhancement by sulfur particles, obtained by alternating gas absorption experiments with H2 S
(for producing the sulfur particles) and (non-reactive) acetylene; shown
are the acetylene experiments. CFe(III); 0 = 100 mol=m3 ; ferric HEDTA,
T = 293 K; Ns = 15:8 1=s.
was measured again from gas absorption experiments without chemical reaction. The value of the absorption rate was
found to have decreased to approximately the original value,
as recorded without the sulfur particles. Small amounts of
surfactant decrease the surface tension which is known to
a*ect adhesion (Vinke et al., 1991). These above experiments thus support the hypothesis that particle-to-interface
adhesion e*ects play a role in the enhancement mechanism.
At elevated pH (pH ¿ 5) the phenomenon of EAexpt
increasing with increasing Fe(III) conversion was not
observed in the stirred cell. However, considerable enhancement of gas absorption rates, possibly resulting from
sulfur precipitates, was reported by Demmink, Wubs, and
Beenackers (1994), who studied H2 S absorption in ferric
NTA solutions in a co-current down-9ow column packed
with static mixers (6:7 6 pH 6 8:3). Therefore, under different hydrodynamic conditions the e*ect of sulfur precipitates on gas absorption may be signi&cant over a wider
pH-range.
To some extent, the above observed role of pH may result
from particle properties, for instance the wettability (which
a*ects adhesion) and the particles’ aJnity for the gaseous
solute. This may be expected because the sorptive capacity
of the sulfur depends upon the electrochemical state of the
surface which is likely to be a*ected by the pH of the reaction media. Fig. 3 shows some results of gas absorption
experiments, batchwise in the gas phase, where alternately
non-reactive gas acetylene and reactive gas H2 S were absorbed in aqueous ferric HEDTA solutions, with pH 4:1; 7:0
and 9:5, respectively. The stirring rate and ionic strength
were kept constant, the latter by adding NaCl. It can be seen
that the particles produced at pH = 4:1 are more e*ective
1727
expt
Fig. 4. EA; p as a function of CS ; values of kL and DA given in Table 2,
T =303 K; pH=4:5. (1) CFe(III); 0 =50 mol=m3 ; (2) CFe(III); 0 =100 mol=m3 ;
(3) CFe(III); 0 = 200 mol=m3 ; (4) CFe(III); 0 = 300 mol=m3 ; (5)
CFe(III); 0 = 400 mol=m3 . For readability, dashed lines are included to connect the observations within one absorption experiment. Curve 2 falling
in between curves 3 and 4 is probably due to experimental error.
than the particles produced at pH = 7:0 and 9:5. No e*ect
of pH on particle size was observed.
An interpolated line through the initial data points, shown
in Fig. 2, line 7, can be used to estimate the e*ect of sulfur
particles on mass transfer, by de&ning
expt
EA;
p
E expt
= Aint ;
EA
(8)
where EAint is the EA -value estimated from line 7 in Fig. 2.
expt
The scaled result, shown in Fig. 4, indicates that EA;
p increases with CS (where CS is the molar sulfur concentration,
back-calculated from the Fe(III)-conversion), which clearly
indicates that an auto-catalytic e*ect is taking place. Also,
at a given CS , the e*ect of sulfur appears to increase with
CFe(III); 0 .
It is therefore suggested that all the sulfur—that already
present in the bulk liquid as well as that produced in a penetration element, during its stay at the gas–liquid interface,
contributes to the enhancement e*ect.
4. Model development
The basic assumptions made in developing the model for
gas absorption with local auto-catalysis are similar to those
made for the particles-adhering-at-interface model reported
by Demmink et al. (1998). Essentially, the gas–liquid interface is considered to be made up of liquid penetration elements which are replaced after a &xed contact time !c , in
accordance with Higbie’s mass transfer theory (Westerterp,
van Swaaij, & Beenackers, 1984). These liquid elements are
partially covered by solid particles, with diameter dp , adhering to the gas–liquid interface. Mass transport takes place
only through the liquid phase for that part of the interface
that is free of particles. In the covered part, the di*usant
moves through a liquid layer and then through a particle
present in its path, &nally di*using into the liquid behind
the particle. The particles have a greater aJnity for the solute compared to the surrounding liquid phase and therefore
surface=internal di*usion is not controlling; these simply act
as reservoirs which receive solute from one side (facing the
gas–liquid interface) and release it from the other (facing
the liquid bulk). The adhering particles are assumed to have
a receiving surface, with area 1=2#d2p , facing the gas–liquid
interface. This receiving surface is located at an average distance from the interface, denoted $p . Facing the liquid bulk,
at average distance of $bp , is the particle’s releasing surface.
As shown in Fig. 5a, the covered liquid phase is thus assumed to consist of two “zones”. Zone I is situated between
the interface and the solid particle, while zone II is on the
back side of the particle, stretching out to the liquid bulk.
Mass balances over both liquid zones, as well as the solid
phase, can be developed and solved, yielding a gas absorption rate for the covered part of the interface, here denoted
JA& . If JA' is the absorption rate through the uncovered part
of the interface, the computed enhancement resulting from
solid particles is given by
comp
EA;
p =
(1 − &)JA' + &JA&
;
JA'
(9)
where, & is the fractional coverage of the interface by the adhering particles. The above description applies to a non-auto
catalytic scenario, wherein the adhering particles are not
produced by an in situ reaction, but added externally, as in
our earlier study (Demmink et al., 1998).
When a surface element from the bulk liquid joins the gas–
liquid interface it contains only particles that it carried with
it from the bulk liquid. However, during its contact time with
the gas phase, solid particles are produced due to reaction
(1) at the interface and therefore the covered surface fraction
increases; see Fig. 5b. The interface now consists of 3 parts:
that covered by particles which arrived from the liquid bulk
and occupy a fraction &, the dynamic fraction (, covered by
particles that are being locally produced after the arrival of
the element at the interface, and the remaining, particle-free
(uncovered) part denoted by the fraction '. Thus,
& + ((t) + '(t) = 1;
(10)
where 0 ¡ t 6 !c . Displacing of “old” particles as a result
of rapid interfacial sulfur precipitation is not included in the
present model. From Eq. (10) it follows that
d'
d(
=−
dt
dt
(11)
with
t = 0;
( = 0;
' = (1 − &):
(12)
1728
Fig. 5. (a) Schematic representation of a single particle adhering to the gas–liquid interface, showing the local concentration gradients. (b) Schematic
representation of growing coverage by adhering solid particles, on the interface of a liquid element at the gas–liquid interface, during a contact time
(
!c . Here, d&p ¿ dp . (b1) Initial coverage: t = 0; ( = 0; & ¿ 0. (b2). Coverage after a time interval t = * (0 ¡ * ¡ !c ): ( ¿ 0, during the time interval
between * and * + d* a new “strip” d( is produced. (b3) Coverage at t = !c .
The fraction & is assumed to be given by a Langmuir-type adhesion isotherm (Demmink et al., 1998; Vinke et al., 1993).
& = &max
m
Keq
Cs
;
mC
1 + Keq
s
(13)
where, Cs is the molar concentration of the particulate material, &max is the maximum possible fractional coverage at
m
a given speed of stirring and Keq
an equilibrium constant.
Since the (-coverage consists of relatively freshly produced particles, the average particle size, denoted by d(p ,
may di*er from the one that makes up the &-coverage (d&p ),
and therefore the average distance from the gas–liquid interface of &- and (-particles, $&p and $(p , respectively, may
also be di*erent (see Fig. 5b). The expressions for the transfer rates through the three surface fractions, using Higbie’s
mass transfer theory, will now be discussed.
4.1. Mass transfer through uncovered surface ('-fraction)
The species balance on A in this part of the interface is
given by
'
'
@2 CA;
@CA;
L
L
'
+ k1 CA;
DA
=
L
@x2
@t
(14)
with the conditions
t = 0;
0 6 x 6 ∞;
t ¿ 0;
x = 0;
'
CA;
L = CA; L = 0;
'
i
CA;
L = CA; L ;
t ¿ 0;
x → ∞;
'
'
CA;
L = CA; L = 0:
(15)
(16)
(17)
The solution to the above equation yields (Danckwerts,
1970)
'
CA;
L (x; t)
i
CA;
L
k1
x
1
x erfc √
− k1 t
= exp −
2
DA
2 DA t
1
+ exp
2
k1
x
x erfc √
+ k1 t
DA
2 DA t
(18)
1729
from which the time-dependent, speci&c rate of absorption
becomes,
' @CA; L
'
JA; i (t) = −DA
@x
x=0
i
= CA;
L
exp(−k1 t)
(DA k1 ) erf ( k1 t) + √
#k1 t
(19)
and this rate, averaged over the contact time !c , therefore
gives
1 !c '
J (t) dt
JA' =
!c 0 A; i
1
exp(−k1 !c)
i
√
= CA;
1
+
erf
(
;
D
k
!
)
+
k
A
1
c
1
L
2k1 !c
#k1 !c
(20)
where !c may be computed from
!c =
4DA
:
#kL2
(21)
4.2. Mass transfer through adhering particles from the
liquid bulk (&-fraction)
For the &-coverage part of the interface, the basic formulation remains identical to the one given in Demmink et al.
(1998). See Fig. 5a for a schematic representation.
Zone I: 0 6 x 6 $&p
I; &
I; &
@2 CA;
@CA;
L
L
I; &
+ k1 CA;
DA
=
L;
@x2
@t
(22)
the conditions being
t=0
0 6 x 6 $&p ;
t ¿ 0;
x = 0;
t ¿ 0;
x=
$&p ;
I; &
CA;
L = CA; L = 0;
I; &
i
CA;
L = CA; L ;
I; &
CA;
L = CA; S =m:
(23)
(24)
(25)
&
Zone II: $b;
p 6x6∞
DA
II; &
II; &
@2 CA;
@CA;
L
L
II; &
+ k1 CA;
=
L
@x2
@t
(26)
t = 0;
&
$b;
p 6 x 6 ∞;
&
x = $b;
p ;
x → ∞;
II; &
CA;
L = CA; L = 0:
(29)
The balance for the accumulation of A within (or on) a
single particle, taken to be a sphere, is given by
&
Zone I
@CA; S
@CA; S
3
−DA
= &
@t
dp
@x x=$&
p
− −DA
@CA; S
@x
Zone II &
&
x=$b;
p
(30)
along with the condition
t = 0;
'
CA; S = mCA;
L = 0:
(31)
For the numerical solution of Eqs. (22) – (25) and (26) –
(29), an implicit, central (&nite) di*erence approximation
was used, in combination with an iterative scheme for the
solution of CA; S from Eqs. (30) and (31).
The speci&c, time-dependent, rate of absorption, for this
type of surface coverage is thus given by
&
@CA; L (t)
&
JA; i (t) = −DA
:
(32)
@x
x=0
As was done before, (Demmink et al., 1998; Vinke et al.,
&
&
1993), $&p = d&p =4 and $b;
p = 3dp =4.
4.3. Mass transfer through freshly precipitated, adhering
particles (( − fraction)
The surface fraction ( increases as a function of the time
spent by the surface element at the gas–liquid interface (t),
by invading the unoccupied surface fraction ' which thus
reduces with time (see Fig. 5b). Hence, for 0 ¡ t 6 !c , the
total (-coverage itself is composed of (“strips” of) particles
that have been produced at di*erent times within this interval. Consider a strip of particles born at time * (i.e., between
time * and * + d*) and let R( = d(=dt be the rate of production of the beta coverage. If JA;( i (*; t) is the time-dependent,
speci&c rate of absorption into a particle strip born at time
*, then a summation of the rate times the strip “thickness”,
over all strips born between * = 0 and the current time * = t
gives
t
((t)JA;( i (t) =
JA;( i (*; t)(R( )t=* d*:
(33)
0
The quantities required for the evaluation of the above integral are derived below.
with
t ¿ 0;
t ¿ 0;
II; &
CA;
L = CA; L = 0;
II; &
CA;
L = CA; S =m;
(27)
(28)
4.3.1. Time-dependent 7ux through a di9erential strip,
born at t = * [JA;( i (*; t)]
The absorption rate JA;( i (*; t) for a (-strip produced at t = *, at any time t is given by the usual
1730
de&nition
JA;( i (*; t) = −DA
along with the condition
I; (
@CA; L
@x
:
(34)
x=0
The equations required for the evaluation of the concentration gradient above may be written by analogy with the
derivations described previously for the & coverage, with a
single exception: any ( particles, tagged by their time of
birth *, occupy previously uncovered area at the interface
'
(' fraction), and therefore “see” a concentration CA;
L (x; *),
given by Eq. (18). The &nal equations may now be given as
Zone I: 0 6 x 6 $(p , time interval: * 6 t ¡ !c
I; (
I; (
@2 CA;
@CA;
L
L
I; (
DA
=
+ k1 CA;
L
@x2
@t
(35)
with conditions
t = *;
I; (
'
CA;
L = CA; L (x; *);
0 6 x 6 $(p ;
t ¿ *;
x = 0;
t ¿ *;
x=
$(p ;
(36)
I; (
i
CA;
L = CA; L ;
(37)
I; (
CA;
L = CA; S =m:
(38)
(
Zone II: $b;
p 6 x 6 ∞, time interval: * 6 t ¡ !c
II; (
II; (
@2 CA;
@CA;
L
L
II; (
+ k1 CA;
DA
=
L
@x2
@t
(39)
with
t = *;
$pb; (
6 x 6 ∞;
t ¿ *;
x=
$pb; ( ;
t ¿ *;
x → ∞;
II; (
'
CA;
L = CA; L (x; *);
(40)
II; (
CA;
L = CA; S =m;
(41)
II; (
'
CA;
L = CA; L = 0:
(42)
Here, by analogy with $p& and $pb; & ; $p( and $pb; ( denote the
average distance of the receiving and releasing surface of the
particle from the interface, respectively, given by $p( = dp( =4,
and $pb; ( = 3dp( =4.
The balance for the accumulation of A within (or on) a
single particle is similar to Eq. (30), and written as

Zone I
(
I; (
@CA;
@CA; S
3 
L
−DA
= (
@t
@x
dp
(
x=$p
II; (
@CA;
L
− −DA
@x
Zone II 

x=$pb; (
(43)
t=*
'
(
CA; S = mCA;
L (x = $p ; t = *):
(44)
The above Eqs. (35) – (44), can now be solved numerically
I; (
II; (
to obtain CA;
L (*; x; t) and CA; L (*; x; t) in the two zones I and
II, respectively, as well as CA; S (*; t). The 9ux de&ned by Eq.
(34) can be evaluated by di*erentiation of the concentration
I; (
pro&le CA;
L (*; x; t).
4.3.2. Rate of formation of ( coverage [R( ]
The ( coverage is formed by the freshly precipitating particles being generated by reaction. Only some of the sulfur
particles, that are “close” enough to the interface, are likely
to adhere to it. Therefore, it is proposed that the particles
precipitated in a certain zone, 0 ¡ x 6 $t , only, and in all
the &; ( and ' areas, have a &nite probability of getting attached to the interface.
Therefore, we may write the implicit equation
$t
d(
i
= Keq
R( =
.P
'(t)k1 CA' (x; t) + &k1 CA& (x; t)
dt
0
t
+
k1 CA( (*; x; t)R( (*) d* d x;
(45)
0
i
where, .P is the stoichiometric factor in A+.B B → .P P; Keq
is a proportionality constant between the amount of product
formed (within the zone 0 ¡ x 6 $t ) and the (-coverage that
is produced by this amount of product; the concentrations,
CA& and CA( denote concentration pro&les whose values may
be set to the appropriate pro&les for zone I or II, depending
upon the domain of integration with respect to x. The terms
on the right-hand side of the above equation give the molar
rate of generation of sulfur due to reaction in the liquid phase
(“behind” each of the &; ( and the ' fractions) in the zone
0 to $t , per unit interfacial area.
i
= 0 denotes the limiting case of new-born
Physically, Keq
particles having no interfacial aJnity. It will be shown bei
low that an upper-bound for Keq
can be determined by geometrical considerations. For $t , two “intuitive” limits are
possible. If only those particles produced at the interface
(between the physical interface and the receiving surface of
the existing, adhering layer) adhere, then $t = $p( ($p( may
be replaced by $p& for sulfur formed in the & coverage). On
the other hand, the limit, $t → ∞ indicates that all locally
produced particles have an equal probability of getting attached to the interface, irrespective of their distance from
the interface. As shown below, for the Ha-values encountered in this work, most particles are produced close to the
interface, and either limiting value of $t may be used.
Since the size of the new-born particles (or nucleates)
is unknown, and only the average particle size in the samples drawn from the bulk of the liquid can be measured,
we set dp( = dp& . This assumption may be considered as an
upper-bound for dp( ; in real systems, it is likely that dp( d&P .
1731
4.4. Overall enhancement by particles
The time-dependent speci&c rate for the entire gas–liquid
interface is given by
JA; i (t) = &JA;& i + ((t)JA;( i (t) + [1 − & − ((t)]JA;' i (t)
(46)
and the surface averaged, speci&c rate follows from the definition
!c
1
JA; i (t) dt;
(47)
JA =
!c 0
Table 3
Model parameters obtained by &tting the model to experimental data in
Fig. 2; kL and DA -values given in Table 2, T = 303 K; pH = 4:5; $t =
$p ; &max = 1
m
m ; m 3 =mol
Keq
i ; m 2 =mol
Keq
k2 ; m3 =mol s
3:5 × 104
3:3 × 10−3
2.4
1.1
the computed enhancement factor, due to the particles, thus
becoming
JA
comp
(48)
EA;
p = ';
JA
where, JA' is given by Eq. (20). Alternatively, the computed
enhancement factor with respect to the maximum physical
absorption rate may be de&ned [analogous to Eq. (4)]
JA
(49)
EAcomp =
i :
kL CA;
L
i
4.5. Estimating an upper-bound for [Keq
]
The proportionality constant between moles of product
i
formed to fractional coverage produced, Keq
, was introduced
in Eq. (45), and its maximum value can be estimated by the
geometric considerations described below.
The maximum value for the integral term on the
right-hand side of Eq. (45) occurs when the penetration element is saturated with A and the concentration everywhere
i
is CA;
L so that the maximum molar generation rate of sulfur
per unit interface area is
i
RS; max = .P k1 CA;
L $t :
(50)
This is equivalent to a number production rate (of particles
of size dp( )
RN; max =
i
.P k1 CA;
L $t Mw; p
p (#=6)(dp( )3
:
(51)
If all the particles adhere to the interface and each particle
blocks an area equal to its projected area then the maximum
fractional-coverage production rate is given by
RN; max
:
(52)
R(; max =
(#=4)(dp( )2
Dividing R(; max by RS; max results in
3Mw; p
i
(Keq
)max =
:
2p dp(
(53)
With dp( = 2 m; Mw; p = 32 kg=kmol and p = 2000 kg=m3 ;
i
(Keq
)max ≈ 12 m2 =mol. Since the values for dp( and p are
i
upper limits, this value of (Keq
)max represents a conservative
estimate.
Fig. 6. Parity plot of experimental and computed enhancement facexpt
comp
tors, EA
and EA ; values of kL and DA given in Table
2, T = 303 K; pH = 4:5; $t = $p ; &max = 1. Regressed values of
model parameters reported in Table 3. (1) CFe(III); 0 = 50 mol=m3 ;
(2) CFe(III); 0 = 100 mol=m3 ; (3) CFe(III); 0 = 200 mol=m3 ; (4)
CFe(III); 0 = 300 mol=m3 ; (5) CFe(III); 0 = 400 mol=m3 .
5. Results and discussion
The parameter values required for using the model presented above were back-calculated from the experimental
data shown in Fig. 2, utilizing a least squares optimization techniques (Press, Flannery, Teukolsky, & Vetterling,
1989). The estimated parameter values are shown in
Table 3. Fig. 6 shows a parity plot of the experimental and
computed enhancement factors. It can be seen that almost
all the experimental data points lie within the 25% variation lines marked on the &gure. Given the complexity of
the phenomena under study, the inherent scatter in the data
on account of somewhat complex experimental procedures,
these model &ts appear to be decent.
i
It may be noted that the value of Keq
determined from the
experimental data is well below the maximum, estimated
from Eq. (53). The value for m is still high, but below the
maximum value predicted by Wubs et al. (1991).
To gain insight into the results as predicted by proposed
model, including its sensitivity to selected parameters, the
comp
values for EA;
p were calculated as a function of CFe(III)
1732
comp
Fig. 7. EA; p and computed values of surface coverage fraction (
at t = !c , both as a function of Ha (varied by changing CFe(III)
m = 3:2 ×
(0 6 CFe(III) 6 450 mol=m3 ). Data used: m = 3:5 × 104 ; Keq
10−3 m3 =mol; k2 =1:1 m3 =mol s; kL =2:3×10−5 m=s; DA =1:4×10−9 m2 ;
(
i = 2:4 m 2 =mol; $ = $ . Lines A1–A6 show the
dp& = dp = 2:0 m; Keq
t
p
e*ect of &-coverage ((A1) & = 0, (A2) & = 0:02, (A3) & = 0:06, (A4)
& = 0:1, (A5) & = 0:5, (A6) & = 1:0). Line A1a ($t → ∞). Line A1b
(
(dp = 0:5 m). Lines A7 and A8 may be compared to line A1 and
i (A7: K i = 1:2 m 2 =mol,
show the model sensitivity to parameter Keq
eq
i
2
A8: Keq = 4:8 m =mol).
(Fig. 7) and kL (Fig. 8). The thin=dotted lines in these &gures refer to the left vertical axis and indicate the values
of the computed enhancement factor under di*erent operating conditions=assumptions (cases A1 to A8). Similarly, the
thick curves pointing to the right vertical axis show the (
surface fraction that has formed on the gas–liquid interface
of a single surface element at t = !c for the various cases.
There are three competing sinks for the uptake of H2 S that
enters a surface element, namely, consumption by chemical
reaction (1), adsorption on & sulfur and uptake by ( sulfur.
The enhancement factor EA; p essentially captures the e*ects
due to uptake of the di*usant by the particles. The eJcacy
of the & particles in picking up the solute tends to diminish
with increasing reaction rates which causes less solute to
remain available for adsorption on the particles. The e*ect
of reaction on the eJcacy of the ( particles is ambiguous
because on the one hand increasing reaction rates increase
the rate of production of the ( coverage, whereas, on the
other, greater reaction rates still lead to lower amounts to be
available for particle uptake. The interplay of these factors
lead to the results shown in Fig. 7. The ( sulfur works at
maximum eJciency in enhancing the absorption rates in the
absence of the & particles as it does not have to compete
with any pre-existing particles.
Line A1 in Fig. 7 shows the intrinsic e*ect of CFe(III); 0 (&=
comp
0) on EA;
p , (keeping DA and kL &xed). Here, the e*ect
of CFe(III); 0 is expressed through the Hatta number, Ha
[see Eq. (7)]. It can be seen that for & = 0, the e*ect of
comp
Fig. 8. EA; p and computed values of surface coverage fraction (
at t = !c , both as a function of kL . Data used: m = 3:5 × 104 ;
m =3:2×10−3 m 3 =mol; K i =2:4 m 2 =mol; k =1:1 m 3 =mol s; D =1:4
Keq
2
A
eq
(
× 10−9 m2 ; dp& = dp = 2:0 m; CFe(III) = 90 mol=m3 . Line B1 & = 0. Line
i set to zero.
B2 & = 0:02. Line B3 & = 0:1. Line B3a & = 0:1 with Keq
comp
increasing ( coverage by reactive production on EA;
p is
very strong. Line ((A1) shows the calculated (-fraction in a
liquid element at t = !c . This coverage increases with Ha, as
more sulfur is produced. For Ha ≈ 15; ( reaches 1 close to
t = !c . Higher values of Ha therefore do not result in growing (-coverage (at t = !c ), whereas the competitive e*ect
of reaction (1), in consuming H2 S increases. This results in
comp
a decreasing trend for EA;
p -values with increasing Ha.
Lines A2, A3, A4, A5 and A6 show the results of similar calculations, with & = 0:02; 0:06; 0:1; 0:5, and 1.0, respectively. Line A2 begins (and remains) at a higher value
than line A1 because of the presence of & particles that were
present in the element when it arrived from the bulk liquid.
The enhancement reduces with increasing Ha as the reaction overcomes the bene&ts of particle uptake. The ( particles are not yet e*ective enough to help but with further
increase in Ha the enhancement due to these ( particles becomes strong enough to raise the absorption rates. Finally,
the limit of maximum surface coverage is reached and the
enhancement factors starts to decline again. A similar explanation applies to the other lines A3 to A6; line A6 has
no room for ( particles and therefore a consistent decline in
the enhancement factor can be seen with increasing Ha.
The dashed line A1a in Fig. 7 is calculated in a manner
similar to A1, however, with $t → ∞. At very low Ha, the
e*ect of ( is too small to be noted, whereas at high Ha,
most sulfur is produced suJciently close to the interface.
Therefore, no di*erence between the limiting cases $t = $p
and $t → ∞ is observed. It thus appears that, under the
circumstances considered here, either assumption may be
used.
1733
The dashed line A1b is also calculated as is A1 but now
with dp( = 0:5 m instead of dp( = (dp& ) = 2 m. It is found
that the assumption made for dp( has a signi&cant e*ect on
comp
the calculated EA;
p .
The dashed lines A7 and A8 are also analogs of A1, but
i
with Keq
set to 1.2 and 4:8 m2 =mol, respectively. The corresponding lines ((A7) and ((A8) show the liquid element’s
(-fraction at t = !c . It is seen that, under the conditions
comp
i
has a large e*ect on EA;
considered here, Keq
p . For high
i
Keq , the maximum (-value is reached at lower Ha, where
the competitive e*ect of reaction (1) is low. As a result, the
comp
maximum EA;
p in line A8 is not only reached at lower Ha,
but is also higher.
comp
Fig. 8 shows the e*ect of kL on EA;
p with CFe(III) &xed
3
at CFe(III) = 90 mol=m . Consider line B1 which is drawn
for & = 0. At low kL , the liquid element’s contact time
(!c ) is very long which results in a large (-coverage but
the eJcacy of the particles in picking up solute is eroded
by reaction which gets so much time to consume the solute. As kL is increased, the enhancement rises on account
of the particles becoming more eJcient in solute uptake
and a faster circulation of the surface elements through
the bulk liquid phase. At some point, with increasing kL ,
the contact time becomes too short for the particles to
adsorb solute and the enhancement begins to fall; hence,
the maximum in the enhancement curves shown in the
&gure.
Lines B2 and B3 were calculated with &nite &-coverage
i.e. &=0:02 and 0.1, respectively. The extra particles present
initially, upon the arrival of the surface element form the
bulk liquid, simply serve to elevate the entire enhancement
i
curve. Line B3a was computed after setting Keq
to zero so
that sulfur particles being produced locally in the penetration
element are not allowed to adhere to the gas–liquid interface.
A comparison with line B3 shows that at large kL the local
e*ects do not have much signi&cance because the contact
times are too small for any substantial local sulfur to get
generated.
The results reported in this study have signi&cance for the
design of industrial processes for removal of H2 S, in which
sulfur is a product. This is especially true of equipment
where the mass transfer coeJcients are of a similar order
of magnitude as those observed in our (typical) laboratory
cell, such as, packed towers, bubble columns, etc.
6. Conclusions
Mw; p
Ns
PA
R
RN
RS
Gas absorption rates in a stirred cell reactor, for the absorption of H2 S into aqueous ferric NTA solutions of low
pH (pH = 4:5), can be enhanced by the precipitated sulfur particles which adhere to the gas–liquid interface. This
auto-catalytic e*ect increases with increasing ferric chelate
concentrations.
A model is presented, based on Higbie’s penetration
theory, which takes into account, both particle-to-interface
adhesion and growing particle coverage during a liquid element’s contact time at the interface. It is shown that the
proposed model is a promising and useful &rst approximation to describing these complex autocatalytic a*ects on the
absorption rates.
Notation
a
CA; L
CA; S
CFe(III)
CFe(II)
CS
dp
DA
EA
EA; p
Ha
He
JA
JA; i
m
Keq
i
Keq
k1
k2
kL
m
R(
t
te
VG
VL
x
z
speci&c gas–liquid interfacial area, m2 =m3 (liquid)
liquid-phase concentration of A, kmol=m3
solid-phase concentration of A, kmol=m3
liquid-phase concentration of ferric chelate,
kmol=m3
liquid-phase concentration of ferrous chelate,
kmol=m3
molar product (sulfur) concentration, kmol=m3
particle diameter, m
liquid phase di*usion coeJcient of A, m2 =s
enhancement factor with respect to maximum
physical rate of absorption, dimensionless
enhancement factor with respect to rate in absence of solids, dimensionless
Hatta number, dimensionless
Henry’s constant, also solubility, kmol=m3 Pa
average speci&c rate of absorption of A,
kmol=m2 s
instantaneous, time-dependent speci&c rate of
absorption of A, kmol=m2 s
Langmuir particle-to-bubble adhesion coeJcient, m3 =kmol
proportionality constant between moles of S
formed to ( coverage produced, m2 =kmol
(pseudo-) &rst-order (liquid phase) reaction rate
constant, 1=s
second-order (liquid phase) reaction rate constant, m3 =kmol s
liquid-phase mass transfer coeJcient, m=s
distribution coeJcient for partitioning of A between solid and liquid phases, dimensionless
molar weight of product, kg=kmol
stirrer speed, 1=s
partial pressure of gas A, Pa
gas constant (=8:314); J=mol K
number production rate of particles, 1=m2 s
molar rate of solids production per unit interface
area, kmol=m2 s
rate of production of ( coverage, 1=s
time, over surface element contact time scale, s
time, over the experiment (clock=batch) time
scale, s
volume of gas space in the reactor, m3
volume of liquid phase, m3
distance from gas–liquid interface, m
ionic charge
1734
Greek letters
&
(
'
$p
$pb
$t
!c
.B ; .P
p
*
interface fractional coverage of adhering particles arriving from the liquid bulk, dimensionless
interface fractional coverage of adhering particles, produced during contact time of penetration element, dimensionless
uncovered interface fraction, dimensionless
“e*ective” distance from the interface of the receiving plane of the adhering particle, m
“e*ective” distance from the interface of the releasing half of the adhering particle, m
penetration depth from within which newly precipitated particles may adhere to the gas–liquid
interface, m
contact time from Higbie’s theory, s
viscosity, Pa s
stoichiometry in A + .B B → .P P, dimensionless
density of reaction liquid, kg=m3
solids (product) density, kg=m3
time of birth of a di*erential ( strip, s
Superscripts
I
II
comp
expt
i
int
&
(
'
pertaining to zone I
pertaining to zone II
computed value
experimental value
value at gas–liquid interface
interpolated value
pertaining to &-covered interface
pertaining to (-covered interface
pertaining to uncovered interface
Subscripts
0
A
A
max
o
initial value with respect to te
pertaining to reactive solute A
pertaining to non-reactive solute A
maximum value
value in pure water
Ligands and surfactants
CTMAB
DTPA
EDTA
HEDTA
NTA
TAMOL
N -cetyl N; N; N -trimethyl ammonium bromide
diethylenetriaminepentaacidic acid
ethylenediaminetetraacetic acid
hydroxyethylethylenediaminetriacetic acid
nitrilotriacetic acid (NTA)
disodium-2-2 -dinaphtyl-methane-6,6 disulfonate
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