CATALYSIS, KINETICS AND REACTORS
Chinese Journal of Chemical Engineering, 19(1) 76—82 (2011)
Adsorption and Ozonation Kinetic Model for Phenolic
Wastewater Treatment*
Wongsarivej Pratarn1,**, Tongprem Pornsiri1, Swasdisevi Thanit2, Charinpanitkul Tawatchai3
and Tanthapanichakoon Wiwut3
1
National Nanotechnology Center, Thailand Science Park, Klong Luang, Pathumthani 12120, Thailand
School of Energy, Environment and Materials, King Mongkut’s University of Technology Thonburi, 126 Pracha
u-tid Rd., Bangkok 10140, Thailand
3
Center of Excellence in Particle Technology, Faculty of Engineering, Chulalongkorn University, Payathai Rd.,
Wangmai, Pathumwan, Bangkok 10330, Thailand
2
Abstract A three phase fluidized bed reactor was used to investigate the combined effect of adsorption and oxidation for phenolic wastewater treatment. Aqueous solutions containing 10 mg·L−1 of phenol and ozone were continuously fed co-currently as upward flow into the reactor at constant flow rate of 2 and 1 L·min−1, respectively. The
phenolic treatment results in seven cases were compared: (a) O3 only, (b) fresh granular activated carbon (GAC), (c)
1st reused GAC, (d) 2nd reused GAC, (e) fresh GAC enhanced with O3, (f) 1st reused GAC enhanced with O3, and
(g) 2nd reused GAC enhanced with O3. The phenolic wastewater was re-circulated through the reactor and its concentration was measured with respect to time. The experimental results revealed that the phenolic degradation using
GAC enhanced with O3 provided the best result. The effect of adsorption by activated carbon was stronger than the
effect of oxidation by ozone. Fresh GAC could adsorb phenol better than reused GAC. All cases of adsorption on
GAC followed the Langmuir isotherm and displayed pseudo second order adsorption kinetics. Finally, a differential
equation for the fluidized bed reactor model was used to describe the phenol concentration with respect to time for
GAC enhanced with O3. The calculated results agree reasonably well with the experimental results.
Keywords adsorption, ozonation, kinetic model, phenol, wastewater
1
INTRODUCTION
Activated carbon, also called activated charcoal
or “activated coal”, is a form of carbon that is made
extremely porous and possesses a very large surface
area available for adsorption and chemical reaction.
Granular activated carbon (GAC) has relatively larger
particle sizes than powdered activated carbon and
consequently has a smaller specific external surface.
Diffusion through the pores of the adsorbate is thus an
important factor. The GAC is nevertheless effective
for adsorption of gases and vapors as their rates of
diffusion are faster than those of liquids. Granulated
carbons are also used for treating water and wastewater to reduce pressure drop and elutriation loss [1].
Ozone (O3) is a triatomic molecule, consisting of
three oxygen atoms. It is an allotrope of oxygen that is
much less stable than the diatomic allotrope (O2).
Ozone in the lower atmosphere is an air pollutant with
harmful effects on the respiratory systems of animals.
Exposure of 0.1×10−6 to 1×10−6 produces headaches,
burning eyes, and irritation to the respiratory passages
[2]. Even low concentrations of ozone in the air could
be destructive to various organic materials such as
latex, plastics, and lungs. Ozone could be generated
by several methods, for example, corona discharge,
ultraviolet light and cold plasma [3]. Industrially,
ozone is used to chemically attack contaminants in
water such as iron, arsenic, hydrogen sulfide, nitrites
and complex organics, often lumped together as color.
In various industrial processes, phenol is one of
the important starting or intermediate materials. Phenol is known to be carcinogenic and possesses high
stability and high toxicity. It has been declared to be
hazardous pollutant even at very low concentration [4].
It can damage the skin and other tissues of the human
and animals. When digested, phenol-containing liquids could lead to liver damage, dark urine and irregular heart beats. Therefore, the treatment of phenolic wastewater is of considerable importance in environmental protection.
Many technologies have been attempted for treating phenolic wastewater, for example, biological
treatment [5], chemical precipitation or oxidation [6],
ion exchange [7], and adsorption [8]. However, there
are few processes to deal with this highly toxic
wastewater with reasonable costs. The degradation of
aqueous phenol by simultaneous use of ozone and
activated carbon or zeolite offers an environmentally
friendly alternative of phenolic treatment [9-13]. In
this research, the removal of phenol by adsorption on
GAC without and enhanced with ozone is investigated
using a laboratory scale three phase fluidized bed reactor. The adsorption isotherm and kinetics of GAC
are examined and modeled. A differential equation to
describe the phenol concentration with respect to time
for the case of GAC enhanced with O3 is derived and
examined.
Received 2010-06-02, accepted 2010-12-27.
* Supported by the National Nanotechnology Center (NANOTEC) (601003) and the National Science and Technology Development Agency (NSTDA).
** To whom correspondence should be addressed. E-mail: [email protected]
Chin. J. Chem. Eng., Vol. 19, No. 1, February 2011
2
2.1
EXPERIMENTAL
Activated carbon preparation
The test GAC made from coconut shell was purchased from Carbokarn (Thailand) Co., Ltd. The GAC
was sieved to select the size range of 0.4-2.0 mm particle diameter. The classified GAC was heated and
held at 473 K for 4 h to eliminate its moisture and
volatile impurities.
2.2
Characterization
77
collected from the effluent stream with respect to time.
The adsorption and oxidation of activated carbon and
ozone were investigated in seven cases: (a) O3 only, (b)
fresh GAC, (c) 1st reused GAC, (d) 2nd reused GAC,
(e) fresh GAC enhanced with O3, (f) 1st reused GAC
enhanced with O3, and (g) 2nd reused GAC enhanced
with O3. Their phenolic elimination performance was
analyzed and evaluated.
2.4
Chemical analysis
Following the Brunauer-Emmett-Teller (BET)
adsorption method, the specific surface area and porosity of the activated carbon were measured via N2
adsorption-desorption isotherms. Test materials were
measured at 77 K using an automatic adsorption apparatus (BELSORP 28, BEL Japan Inc.). The morphological structure of the activated carbon was characterized by scanning electron microscope (SEM, Hitachi S-3400N).
The progress of the removal was followed by periodically taking the liquid samples from the reactor
for immediate analysis after filtration through 0.45 μm
nylon filter. Phenol was identified and quantified by
high performance liquid chromatography (HPLC,
Shimadzu LC-20A Series) with a diode array detector
at wavelengths of 210 and 254 nm. A 5 μm column of
C18 (Inertsil ODS-3, 25 cm long, 4.6 mm diameter)
was used as stationary phase and a mixture of 4
mmol·L−1 aqueous sulphuric and acetonitrile (volume
ratio of 4︰1) was used as mobile phase at 1 ml·min−1.
2.3 Apparatus and procedure
3
A laboratory scale three phase fluidized bed reactor was used as the adsorption and oxidation system
to improve mixing and homogeneity. Its schematic
diagram is shown in Fig. 1. The reactor with an effective volume of 272 ml was made from transparent
acrylic that allowed the observation of the phenomena
inside. The outside diameter and height of the reactor
were 40 and 300 mm, respectively. The aqueous solution containing 10 mg·L−1 of phenol was treated and
256.8 g·h−1 of O3 was used as oxidizing agent. A constant flow rate of 1 L·min−1 of the solution and 2 L·min−1
of O3 were continuously fed co-currently into the reactor as upward flow. Then, the liquid effluent stream
was recycled to the hold-up tank. 6 L of phenolic
wastewater containing in the tank of 15 L was treated
with activated carbon of 5 g. The solution temperature
in the tank was monitored and controlled at 303 K
using a thermocouple and a cooler. The samples were
RESULTS AND DISCUSSION
3.1
Characterization of activated carbon
The pore size distribution of GAC measured with
the MP-Plot method was found to be essentially microporous with a modal peak of 0.6 nm, and the BET
surface area and total pore volume of GAC were 1154
m2·g−1 and 0.49 cm3·g−1, respectively. To understand
the morphology of GAC surface, its image was obtained with scanning electron microscope (see Fig. 2)
to reveal the high surface area structure of GAC. The
abundant micropores provide superb condition for the
adsorbed material to interact with the ozone at high
concentration.
Figure 2
3.2
Figure 1 Schematic diagram of experimental apparatus
1—hold-up tank; 2—liquid pump; 3—ball valve; 4—liquid flow
meter; 5—air flow meter; 6—ozone generator; 7—air pump; 8—
three phase fluidized bed reactor
Scanning electron microscope image of GAC
Adsorption isotherms
The capacity of virgin activated carbon to adsorb
aqueous phenol was determined by measuring the adsorption isotherm at 298 K, as shown in Fig. 3. The
78
Chin. J. Chem. Eng., Vol. 19, No. 1, February 2011
The Langmuir and Freundlich parameters for the
adsorption of phenol onto activated carbon are listed
in Table 1. This table and Fig. 3 show conclusively
that Langmuir isotherm provides a significantly higher
regression coefficient R2 than Freundlich isotherm.
This means that the surface of GAC is made up of
homogeneous adsorption patches, which is in good
agreement with other reports [21-23].
Figure 3 Equilibrium adsorption isotherm of fresh activated carbon fitted with Langmuir and Freundlich adsorption isotherm
○ fresh GAC;
Langmuir isotherm;
Freundlich isotherm
analysis of the isotherm data is important to develop
an equation that accurately represents the equilibrium
and can be used for design purpose. In this study, two
commonly used equilibrium models are selected to
describe the adsorption data, namely the Langmuir
and the Freundlich isotherm equations [14-20].
The Langmuir theory assumes a homogeneous
monolayer adsorption within the adsorbent. The linear
form of the Langmuir isotherm equation is represented
as
q K C
qe = max L e
(1)
1 + K L Ce
After rearrangement Eq. (1) becomes
Ce
1
1
=
Ce +
qe qmax
K L qmax
Adsorption isotherm constants for the phenol onto
virgin activated carbon at 30 °C
Langmuir model
Freundlich model
Adsorbent
qmax
/mg·g−1
KL
/L·mg−1
R2
n
KF
/L·g−1
R2
fresh GAC
232.558
0.149
0.997
3.504
50.096
0.974
The GAC adsorption of phenol as a function of
time is shown in Fig. 4. Obviously fresh GAC presents slightly faster adsorption than 1st reused GAC
though their final capacities are essentially the same.
This implies that the regeneration is complete though
some of the micropores become slightly narrower.
Similarly, 1st reused GAC presents slightly faster adsorption than 2nd reused GAC. After 360 min, the
adsorption equilibrium is reached. The adsorption of
both GAC and reused GAC follow the pseudo second
t
1
t
=
+ .
order kinetics, that is
2
qt ka 2 qe qe
(2)
where qe is the equilibrium phenolic concentration on
the adsorbent (mg·g−1), Ce the equilibrium concentration of phenol in the solution (mg·L−1), qmax the
maximum monolayer adsorption capacity of the adsorbent (mg·g−1), and KL is the Langmuir isotherm
constant (L·mg−1) related to the free energy of adsorption. A plot of Ce/qe versus Ce for the adsorption of
phenol onto the activated carbon gives a straight line
of slope of 1/qmax and intercept of 1/qmaxKL.
The Freundlich isotherm is an empirical equation
assuming that the adsorption takes place on heterogeneous surfaces of solids and in multilayer sorption
manner. The adsorption capacity is the adsorbed concentration of phenol at equilibrium. The linear form of
Freundlich equation is
qe = K FCe1/ n
Table 1
(3)
After rearrangement Eq. (3) becomes
1
ln qe = ln Ce + ln K F
(4)
n
where KF (L·g−1) and n are Freundlich adsorption isotherm constants indicative of the adsorption capacity
and the degree of nonlinearity between solution concentration and adsorption, respectively. The plot of
lnqe versus lnCe is employed to generate KF value
from the intercept and n value from the slope.
Figure 4 GAC adsorption of phenol as a function of time
fresh GAC; △ 1st reused GAC; □ 2nd reused GAC
○
3.3
Adsorption kinetics
Two kinetic models, the pseudo first order and
pseudo second order, were tested with the experimental data to elucidate the adsorption phenomenon with
the air fed co-currently instead ozone. In the case of
the first order rate equation for GAC, with 1st reused
GAC and 2nd reused GAC, the values of ka1 and qe
were calculated from the slope and intercept of the
plot of lg(qe−qt) versus t (see Fig. 5) and summarized
in Table 2. It is found that the correlation coefficients
for the pseudo first order model are significantly lower
than those of the pseudo second order model. Coupled
with the fact that the lines in Fig. 5 are not straight, it
conclusively shows that the adsorption process does
not follow the first order kinetics.
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Chin. J. Chem. Eng., Vol. 19, No. 1, February 2011
Table 2
Experimental and calculated adsorption ability and rate constant of GAC with 1st reused GAC and 2nd reused GAC
Sample
Pseudo-first order
Pseudo-second order
qe (exp)/mg·g−1
qe (cal)/mg·g−1
kal (exp)/mg·g−1·min−1
R2
qe (cal)/mg·g−1
ka2/g·mg−1·min−1
R2
fresh GAC
11.90
7.36
0.0157
0.9551
12.69
0.0038
0.9994
1st reused GAC
11.89
7.76
0.0150
0.9497
12.94
0.0029
0.9977
2nd reused GAC
11.86
8.19
0.0150
0.9635
13.00
0.0026
0.9976
Figure 5 Pseudo first order analysis for GAC with 1st
reused GAC and 2nd reused GAC
○ fresh GAC; △ 1st reused GAC; □ 2nd reused GAC
In the case of the second order rate equation for
GAC, with 1st reused GAC and 2nd reused GAC, the
values of ka2 and qe were calculated from the plot of
t/qt against t (see Fig. 6 and Table 2). The calculated qe
values agree essentially with the experimental values.
The correlation coefficients for the pseudo second
order kinetic plots are very high, indicating that the
adsorption kinetics of both GAC and reused GAC are
the pseudo second order.
Figure 6 Pseudo second order analysis for GAC with 1st
reused GAC and 2nd reused GAC
○ fresh GAC; △ 1st reused GAC; □ 2nd reused GAC
3.4
Ozonation kinetics
The transient of phenolic degradation with O3
only (ozone-rich air without GAC) was analyzed to
determine the rate constant kr for pseudo first order
kinetics. The rate constant was determined from the
slope of −ln(C/C0) vs. t by fitting the data at 0-180
min, as shown in Fig. 7. Here C0 and C are the phenol
concentration at the initial and at time t, respectively.
The experimental results of the degradation of phenol
solution reveals that the employment of O3 only gives
an initial rate constant kr = 0.0122 L·g−1·min−1 with R2
value of 0.9907. In this case the degradation of phenol
follows the pseudo first order kinetics since the plot
of −ln(C/C0) vs. t presents a straight line.
Figure 7 The plot of −ln(C/C0) vs. t for the reaction between phenol and ozone
3.5
The coupling between adsorption and ozonation
In three out of the seven experimental cases, the
coupled adsorption and oxidation performance of
GAC and ozone was investigated under the same experimental conditions. The three cases are (e) fresh
GAC enhanced with O3, (f) 1st reused GAC enhanced
with O3, and (g) 2nd reused GAC enhanced with O3.
The results for the seven cases are presented in Fig. 8.
Each of the shown data points is the average value of
triplicate data. As expected, using only O3 is the worst
for phenol degradation. When employing only GAC
and air, the degradation performance is significantly
better. In the cases of GAC enhanced with O3, the
performance is further improved significantly. In the
best case, complete degradation of phenol is achieved
within 75 min. In summary, the removal effect of adsorption by GAC is stronger than that of oxidation by
ozone. GAC adsorbs phenolic compounds in aqueous
solutions until it reaches equilibrium. Interestingly,
when coupling the effect of adsorption together with
the effect of ozonation, phenol can be eliminated
faster than in the four previous cases, either O3 or
GAC only.
Figure 8 Phenol concentration as a function of time for
fresh and reused GAC without and enhanced with O3
× O3 only; ○ fresh GAC-air; △ 1st reused GAC-air; □ 2nd reused
GAC-air; ● fresh GAC-O3; ▲ 1st reused GAC-O3; ■ 2nd reused GAC-O3
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Chin. J. Chem. Eng., Vol. 19, No. 1, February 2011
In Fig. 8, the phenol adsorption of GAC in the
three cases without ozone, fresh GAC and 1st reused
GAC is the fastest and second fastest, respectively.
This is consistent with the kinetic results shown in
Fig. 4. Similarly, the phenol elimination in the three
cases of GAC enhanced with O3, fresh GAC enhanced
with O3 and 1st reused GAC enhanced with O3, is the
best and second best, respectively.
However, in actual wastewater treatment, using
fresh activated carbon in each batch is economically
impractical. The synergistic effect of adsorption with
GAC and ozonation is an attractive alternative. Fortunately, the employment of used GAC with or without
ozone shows small difference in adsorption rates after
two regenerations.
3.6
Adsorption and ozonation kinetics
written as
dqt
dC
+V A
(11)
dt
dt
where mc is catalyst loading and V is liquid volume in
the reactor. Substitution of Eqs. (6) and (10) into Eq.
(11) yields
dC
2
(12)
mc ka 2 ( qe − qt ) + V A + mc kr CA = 0
dt
To integrate this equation, it is first written as an ordinary differential equation as follows
M ( t , CA ) ⋅ dt + N ( t , CA ) ⋅ dCA = 0
(13)
−rA mc = mc
More specifically,
2
⎡
⎤
⎣ mc ka 2 ( qe − qt ) + mc kr CA ⎦ ⋅ dt + V ⋅ dCA = 0 (14)
where
M ( t , CA ) = mc ka 2 ( qe − qt ) + mc kr CA
(15)
N ( t , CA ) = V
(16)
2
The ozone oxidation data were analyzed with the
following second order rate equation for irreversible
bimolecular reaction:
dC
−rA = − A = kr CA CB
(5)
dt
where kr is reaction rate constant, CA and CB are phenol and ozone concentrations, respectively. In our experiments, the concentration of ozone was replenished
continuously and was essentially constant, so the rate
equation can be reduced to first-order.
dC
−rA = − A = kr CA
(6)
dt
With separation of variables and integration of both
sides, Eq. (6) becomes
C
− ln A = kr t
(7)
CA0
where CA0 is the initial concentration of phenol.
The adsorption data could be analyzed with either
pseudo first order or pseudo second order kinetics:
kt
lg ( qe − qt ) = lg qe − a
(8)
2.303
t
1
t
=
+
2
qt ka 2 qe qe
(9)
where qe and qt are adsorption ability at equilibrium
and at time t (mg·g−1) respectively, ka1 and ka2 are the
first and second order adsorption rate constants, respectively. Our conclusion in Section 3.3 indicates a
pseudo second order kinetics. Differentiation of both
sides of Eq. (9) gives
dqt
2
= ka 2 ( qe − qt )
(10)
dt
Based on the assumptions of complete mixing in
the fluidized bed and simultaneous adsorption and
ozonation in the reactor, the mass balance equation for
phenol in the three phase fluidized bed reactor can be
and
It should be noted that
∂M
∂N
= mc kr ≠
=0
∂CA
∂t
(17)
Eq. (12) is not an exact ordinary differential equation,
but it becomes an ODE after multiplication by an integrating factor, μ,
⎡ ⎛ ∂M ∂N ⎞ 1 ⎤
μ = exp ⎢ ∫ ⎜
−
⎟ ⋅ dt ⎥
⎣ ⎝ ∂CA ∂t ⎠ N ⎦
⎛ mk
⎞
⎛m k ⎞
= exp ⎜ ∫ c r dt ⎟ = exp ⎜ c r t ⎟
⎝ V
⎠
⎝ V ⎠
Then Eq. (12) becomes
d⎡
⎛ m k ⎞⎤
V ⎢CA exp ⎜ c r t ⎟ ⎥
dt ⎣
⎝ V ⎠⎦
⎛m k ⎞
2
= − mc ka 2 ( qe − qt ) exp ⎜ c r t ⎟
⎝ V ⎠
The integration of Eq. (19) gives
2
⎛m k ⎞
⎛m k
VCA exp ⎜ c r t ⎟ = − mc ka 2 ∫ ( qe − qt ) exp ⎜ c r
⎝ V ⎠
⎝ V
and Eq. (10) can be integrated to yield
1
1
= + ka 2 t
qe − qt qe
or
qe − qt =
qe
1 + qe ka 2 t
(18)
(19)
⎞
t ⎟ dt
⎠
(20)
(21)
(22)
Substitution of Eqs. (21) and (22) in Eq. (20) gives
m k q2
⎛ mk
CA = − c a 2 e exp ⎜ − c r
V
⎝ V
⎛m k ⎞
exp ⎜ c r t ⎟
⎞
⎝ V ⎠ dt (23)
t⎟∫
⎠ (1 + qe ka 2 t )2
Chin. J. Chem. Eng., Vol. 19, No. 1, February 2011
Since the integral in Eq. (23) can not be integrated analytically, it is integrated using Wolfram
Mathematica Software. To obtain some approximate
analytical solution, the following indefinite integral is
utilized
⎛m k ⎞
exp ⎜ c r t ⎟
⎛ kr mc
⎝ V ⎠
∫ (1 + q k t )2 dt = exp ⎜⎝ − ka 2 qeV
e a2
⎞
⎟⋅
⎠
⎡ kr mc ( ka 2 qe t + 1) ⎤
⎪⎧
⎨kr mc ( ka 2 qe t + 1) ⋅ Ei ⎢
⎥−
ka 2 qeV
⎪⎩
⎣
⎦
⎡ k m ( k q t + 1) ⎤ ⎫⎪
ka 2 qeV exp ⎢ r c a 2 e
⎥⎬
ka 2 qeV
⎣
⎦ ⎪⎭
⎡⎣ ka22 qe2 ( ka 2 qeVt + V ) ⎤⎦ + Const
(24)
Application of the initial condition CA = CA0 at t = 0
allows the evaluation of the Const term.
Let us evaluate the complex exponential
⎡ k m ( k q t + 1) ⎤
integral term, Ei ⎢ r c a 2 e
⎥ , by setting z =
ka 2 qeV
⎣
⎦
kr mc ( ka 2 qe t + 1)
and making use of the Puiseux seka 2 qeV
ries of Ei(z) along the positive real axis,
Ei ( z ) = γ + ln z + z +
1 2 1 3
z + z +
4
18
1 4
1 5
z +
z + ⋅⋅⋅
(25)
96
600
where γ is Euler gamma constant, γ = 0.577216···. Substitution of Eqs. (24) and (25) in Eq. (23) gives the
phenol concentration CA with respect to time t analytically as an infinite series.
Figure 9 shows the aqueous phenol concentration
as a function of time for the three cases of GAC enhanced with O3. The comparison shows that the cal-
81
used to predict the phenol concentration with respect
to time and to design a three phase fluidized bed reactor for wastewater treatment.
4
CONCLUSIONS
Removal of phenol in a laboratory scale three
phase fluidized bed reactor with activated carbon either without or enhanced with ozone was examined
experimentally and theoretically. Phenolic degradation
using only O3 gave the worst result among the seven
cases while using the coupling effect of GAC and O3
provided the best result. As expected, the phenol removal performance of fresh and 1st reused GAC
turned out best and second best, respectively. This
holds true for the cases of GAC without and GAC
enhanced with ozone. The oxidation of phenol in the
case of only O3 followed pseudo first order kinetics.
Adsorption of phenol in all cases of GAC followed
Langmuir isotherm and pseudo second order kinetics.
Finally, the derived differential equations and their
solutions for the three phase fluidized bed reactor
yielded predicted results that agreed reasonably with
the corresponding experimental results.
NOMENCLATURE
C
CA
CB
Ce
Ei
KF
KL
ka1
ka2
kr
mc
n
qe
qmax
qt
rA
t
V
z
γ
μ
concentration, mg·L−1
phenol concentration, mg·L−1
ozone concentration, mg·L−1
concentration of phenol in the solution, mg·L−1
exponential integral
Freundlich isotherm constants, L·g−1
Langmuir isotherm constant, L·mg−1
first order adsorption rate constant, mg·g−1·min−1
second order adsorption rate constant, g·mg−1·min−1
first order apparent kinetic rate constant, L·g−1·min−1
catalyst loading, g
degree of nonlinearity between solution concentration and adsorption
adsorption ability at equilibrium, mg·g−1
maximum monolayer adsorption capacity of adsorbent, mg·g−1
adsorption ability at time t, mg·g−1
oxidation reaction rate of phenol, mg·g−1·min−1
time, min
liquid volume, L
mathematical term
Euler gamma constant
integrating factor
Subscripts
Figure 9 Comparison between the calculated CA (line)
and experimental CA (dot) for GAC enhanced with O3
exp.: ○ fresh GAC-O3; △ 1st reused GAC-O3; □ 2nd reused
GAC-O3;
cal.:
fresh GAC-O3;
1st reused GAC-O3;
2nd
reused GAC-O3
culated values of CA from the differential equations
agree reasonably well with the corresponding experimental results. Therefore, both the computational results and the approximate analytical solution can be
A
B
r
0
phenol
ozone
reaction
initial condition
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