Environ. Sci. Technol. 2007, 41, 2908-2914
Photocatalytic Degradation of Two Volatile Fatty Acids in an Annular Plug-Flow Reactor; Kinetic Modeling and Contribution of Mass Transfer Rate P I E R R E - F R A N C¸ O I S B I A R D , ABDELKRIM BOUZAZA,* AND DOMINIQUE WOLBERT Sciences Chimiques de Rennes, UMR 6226, CNRS-ENSCR, EÄ quipe Chimie et Inge´nierie des Proce´de´s, EÄ cole Nationale Supe´rieure de Chimie de Rennes, avenue du Ge´ne´ral Leclerc, 35700 Rennes, France
This study investigates the influence of inlet concentration and of flow rate on the degradation rate of two Volatile Fatty Acids (butyric and propionic acids). TiO2-coated nonwoven fiber textile was used as the photocatalyst in an annular plug-flow reactor at laminar flow regime. The kinetic follows a Langmuir-Hinshelwood form. The oxidation rate increased with the flow rate, which emphasizes the influence of the mass transfer. A first design equation is proposed considering that the mass transfer could be neglected. Despite a good accuracy of the model, the determined kinetic constants are dependent on the flow rate which highlights the contribution of the mass transfer rate on the global degradation rate. Thus, a new design equation which includes the mass transfer rate was developed. Using this model, the degradation rate can be determined for any given flow rate. Moreover, it allows the estimation of the contribution of mass transfer and chemical reaction steps at given experimental conditions; and thus providing an interesting tool for reactor optimization or design.
Introduction Industrial, agricultural, and domestic activities are responsible for the emission of many volatile organic compounds (VOCs) (1). These compounds have drawn considerable attention in the past decade as they are major contributors to air pollution (2). The existing processes of emission control (incineration, absorption, adsorption, condensation, and biofiltration) are not entirely satisfactory (3). Since a large number of VOCs are oxidizable, their photocatalytic oxidation (PCO) appears to be a promising process for their remediation (2). At moderate conditions (room temperature, atmospheric pressure, and with molecular oxygen as the only oxidant), the photocatalysis can degrade a broad range of pollutants into innocuous final products such as CO2 and H2O (4, 5). This technique involves photoreactions which occur at the surface of a catalyst which concern, for the most part, titanium dioxide (TiO2). The reactive species (hydroxyl radical HO• as the primary oxidant and HO•2, RO•2 , O-• 2 ) are created after irradiation of TiO2 by photons of greater energy than the semiconductor band gap * Corresponding author phone: +33 2 23 23 8056; fax: +33 2 23 23 8120; e-mail:
[email protected]. 2908
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(i.e., in the UV domain, for λ < 388 nm). Several excellent reviews are focused on the principle, the mechanisms, and the application of the photocatalysis (6-10). The removal of the contaminant can be described by three main steps in series: first, it transfers from the bulk to the surface of the catalyst (external and internal mass transfer); second, it adsorbs on the surface, and finally it is destroyed by the reactive species on the active sites of the surface. Consequently, the three most important parameters governing the performance of a photocatalytic reactor are the VOC convective mass transfer, the reaction rate, and the illuminated reaction surface area (11). Unfortunately, these three parameters are greatly dependent on many factors. The illuminated reaction surface area and the VOC convective mass transfer are dependent on catalyst configuration and reactor geometry (12-14). The VOC convective mass transfer is also influenced by the physical properties of the contaminant and by flow rate. The degradation rate depends on the physical and chemical properties of the contaminant, UV intensity, temperature, humidity level and contaminants, oxygen, and byproducts concentrations (15-18). Consequently, the performance is directly dependent on the effluent and reactor configuration, which implies that the design and scale up of photocatalytic treatment plant are complex problems. Thus, there is a great need of simple kinetic models. In most of the cases, steady-state kinetic models presented in the literature ignore the influence of mass transfer. This is not a problem if the flow rate is sufficiently important to ensure a good mass transfer rate (3, 16). Nevertheless, working with high flow rates is not usually possible, and then, the observed degradation rate is dependent on mass transfer rate and, consequently, on flow rate. Several authors have developed simple and easy to use kinetic models which take mass transfer into account. They always consider a first-order kinetic which is generally true at low contaminant concentration (12, 13, 19, 20). Nevertheless, at higher concentrations, this assumption cannot be done as zero-order kinetics are observed. The fixed-bed reactor, also called the thin film reactor, is one of the most commonly used due to its ease of operation (21). With this configuration, the external mass transfer has a significant influence (12, 13). Our investigations have been conducted in an annular plug-flow reactor in a thin film configuration. This study is focused on the PCO of two malodorous volatile fatty acids (VFAs): propionic and butyric acids. To investigate the effect of inlet concentration and of flow rate on degradation rate, O2 concentration, humidity level, UV intensity, and temperature are kept constant. The O2 consumption in the reactor is assumed to be negligible as the amount of O2 is really bigger than the amount of photons at the surface of the catalyst. The first aim was to demonstrate that flow rate has a strong influence on degradation rate and should be taken into account in design equations. The second aim was to propose a new design equation based on Langmuir-Hinshelwood and mass transfer models which would allow the degradation rate to be determined whatever the flow rate. This design equation enables the determination of the respective contribution of mass transfer and reaction steps, which is useful for the optimization of the reactor.
Experimental Section Experimental Setup. The experimental setup is presented Figure 1. The ambient air is the gas carrier. The air stream is generated by a fan. A flow meter (Bronkhorst IN-FLOW) allows the flow rate to be controlled. A variable part of the 10.1021/es062368n CCC: $37.00
2007 American Chemical Society Published on Web 03/17/2007
TABLE 1. Values for the Reynolds Number (Re), Superficial Velocity (u), Mass Transfer Coefficients (km), and Residence Time in the Catalytic Zone for Each Flow Rate (Q) at 30 °Ca Q u (Nm3‚h-1) (m‚s-1) 2 4 6
flow is derived through a packed column where water flows in counter current in order to humidify the gas at 50 ( 5% RH. This RH is optimal for the VFAs degradation (22). A heating tape, placed upstream to the injection zone, facilitates the volatilization of the pollutants. The VFAs are injected continuously in liquid form by a syringe/syringe driver combination. The temperature in the injection zone is approximately 45-50 °C. A static mixer allows the effluent upstream to the photoreactor to be homogenized. The temperature in the photoreactor is 31 ( 2 °C. Two septa downstream and upstream of the reactor allows outlet and inlet gas to be sampled with a 500 µL syringe. Reactor Elements. The annular plug-flow reactor is composed of two concentric cylinders (Ø 58 and 76 mm) (Figure 2). The effluent flows between them. The total length of the reactor is 1.5 m but the length of the test zone is 1 m. A fluorescent lamp (TL 80 W Cleo Performance provided by Philips) is placed in the smaller diameter cylinder. This lamp has a wavelength spectrum of 310-400 nm with a maximum peak emission at 365 nm. The photocatalyst medium used was provided by Ahlstro¨m. It consists of a nonwoven perforated and porous fiber textile on which 25 g of catalyst (TiO2 Millenium PC500, specific area ) 317 m2‚g-1) are coated per m2. This support has a thickness of 250 µm and is deposited in the inner surface of the outer wall over a length of 0.8 m for a total area of 0.19 m2. Thus the total effective catalyst area per unit volume of the reactor (av) is 126 m2‚m-3. The incident light intensity at 365 nm at the surface of the catalyst is approximately 40 W‚m-2 (measured by a radiometer Vilber Loumat VLX 3W). Experimental Procedures. First, the gas flows and humidity level are set. Then, the pollutants are injected at the adequate rate to obtain the desired inlet concentration. A transient period ensues as the pollutant and the water adsorb onto the titanium dioxide. After the adsorption process reaches equilibrium as indicated by equality between the inlet/outlet pollutant concentrations, the UV lamp is turned on. With the UV lamp on, a transient period precedes a steady-state phase that was established when the outlet pollutant concentration reaches a steady state. The difference between the inlet and outlet concentration allowed the calculation of the oxidation rate (r in µmol‚ m-2‚s-1) and conversion percentage (η) defined as follows: out Q × (Cin PA - CPA ) and rBA ) S out Q × (Cin BA - CBA ) 1000 (1) S out Cin PA - CPA
ηPA ) 100
Cin PA
km,AP (m‚s-1)
km,AB (m‚s-1)
525 2.6 × 10-3 2.8 × 10-3 1050 3.8 × 10-3 4.1 × 10-3 1580 4.8 × 10-3 5.2 × 10-3
residence time (s) 2.5 1.2 0.8
a The mass transfer coefficients are calculated using semiempirical correlation for an annular reactor. This aspect will be discussed later.
FIGURE 1. Experimental setup.
rPA ) 1000
0.33 0.65 0.98
Re
out Cin BA - CBA
and ηBA ) 100
Cin BA
(2)
where Cin and Cοut are, respectively, the inlet and outlet contaminant concentration (mmol‚m-3), Q is the volumetric flow rate (m3‚s-1), and S is the surface of the catalytic support
(m2). The subscripts PA and BA refer, respectively, to propionic acid and butyric acid. For each pollutant, three flow rates were tested leading to a laminar regime in the reactor (Table 1). Control experiments were carried out without catalyst but with UV lamp and show equality between the inlet and outlet concentrations of the VFA which emphasizes the fact that the photolysis is negligible. Analytical Method. Samples of 500 µL were directly injected into a gas chromatograph (Fisons GC 9000 series) for the measurement of the concentration of the PA and BA. A 25 m length capillary column Chrompact FFAP-CB, specially adapted to VFAs, allows the separation of the products and byproducts. A flame ionization detector (FID) was used for detection, with nitrogen as the carrier gas. The temperature conditions for the oven, the injection chamber, and the detector were, respectively, 115, 200, and 250 °C.
Results and Discussion The photocatalytic oxidation of BA and PA was studied under constant humidity, O2 concentration, and UV intensity in an annular plug-flow reactor. The influence on the degradation rate of the inlet concentration of the pollutants and of flow rate is investigated before trying to model the results. Effects of the Inlet Concentration and Flow Rate on the Degradation Rate. For the two pollutants, the degradation rate increases with the inlet concentration (Figure 3). At low inlet concentration, the degradation obeys first-order kinetics, i.e., the degradation rate is proportional to the inlet concentration. In this case, the limiting steps are the adsorption or the transfer of the pollutant onto the catalyst. Not all the photoactive sites are occupied and an augmentation of the bulk concentration results in a higher surface concentration which increases the degradation rate. At higher inlet concentration, the degradation rate tends toward a limit. In this case, the active sites are saturated and an augmentation of the concentration does not result in an increase of the degradation rate. The limitation is due to the turnover or the creation of the active sites. Concerning the effect of the flow rate, we note that when this one increases, the degradation rate increases also. This result emphasizes the influence of the mass transfer at laminar regime. An augmentation of the flow rate results in a higher mass transfer and a smaller concentration gradient between the bulk and the catalyst surface. The flow rate range between surface reaction controlling and diffusion controlling condition is the transitional condition (23). Under transitional condition, the apparent oxidation rate depends on both the surface reaction and gas-phase mass transfer rates. We note that the conversion decreases when the inlet concentration increases (Figure 4). For a given flow rate, the augmentation of the degradation rate cannot compensate the increase in the amount to be treated when the inlet concentration increases. When the flow rate increases, two antagonistic effects are brought into play: the decrease in the residence time in the photocatalytic reactor and the increase in the mass transfer VOL. 41, NO. 8, 2007 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 2. Scheme (a) and sectional drawing (b) of the annular plug-flow reactor.
FIGURE 3. Evolution of the degradation rate vs the inlet concentration of PA (case a) and BA (case b) for various flow rates. T ) 30 °C, RH ) 50%, I ) 40 W‚m-2. The curves correspond to the values predicted by Model I.
FIGURE 4. Evolution of the degradation percentage versus the inlet concentration of PA (case a) and of BA (case b) for various flow rates. T ) 30 °C, RH ) 50%, I ) 40 W‚m-2. rate. As there is a diminution of the conversion with the flow rate (Figure 4), the decrease in the residence time is the prevalent factor. Usually, the studies which investigate the 2910
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effect of the flow rate are carried out in a recirculation batch system. In this case, the degradation rate is dependent only on the mass transfer rate, which implies that better degrada-
TABLE 2. Values of the Langmuir-Hinshelwood Constants for Model I. T ) 30 °C, RH ) 50%, I ) 40 W.m-2a propionic acid
a
flow rate (Nm3‚h-1)
kPA (mmol‚m-3.s-1)
KPA (m3‚mmol-1)
2 4 6
2.10 2.07 2.02
0.32 0.35 0.41
butyric acid
R2 (%)
kBA (mmol‚m-3‚s-1)
KBA (m3‚mmol-1)
R2 (%)
>99.99 99.99 >99.99
2.25 2.16 1.98
0.24 0.27 0.39
99.98 99.99 99.99
The correlation coefficient refers to the outlet concentration.
FIGURE 5. Linear regression of the experimental results with eq 7 for PA (graph a) and for BA (graph b) for various flow rates (2, 4, and 6 Nm3‚h-1). tion rates are obtained when the flow rate increases (13, 14). The two acids have the same behavior. Propionic acid seems to be slightly better degraded than butyric acid even if the latter is more volatile and should have a better affinity for the catalyst. The aim of the next part is to propose a model to describe the observed kinetics. To simplify the problem resolution, we consider only the surface reaction step, i.e., the mass transfer is neglected. Model I: The Mass Transfer Step is Negligible. As the reaction is apparently first order at low reactant concentration and zero order at high values, the assumed rate form is that known as Langmuir-Hinshelwood (24).
KC ∂C u ) -r ) -k ∂z 1 + KC
(4)
For a plug-flow reactor, the design equation is
∫
Cout
Cin
1 + KC dC ) kKC
u
∫
L
-dz
(5)
0
Integration of this equation leads to
[
u
]
1 Cout - Cin + (ln Cout - ln Cin) ) -L k kK
(6)
Rearranging eq 6:
KC r)k 1 + KC
(3)
Where r is the degradation rate (Μ‚m-3‚s-1 or in M‚m-2 of catalyst‚s-1), K is the adsorption constant (in m3‚mmol-1), k is the reaction constant (with the same unity as the degradation rate), and C is the concentration of the contaminant (in mmol‚m-3). This model assumes that the adsorption obeys to the model of Langmuir, that the rate of adsorption of the substrate is greater that the rate of any subsequent chemical reactions, and that no irreversible blocking of active sites by binding to products occurs (7). k is a measure of intrinsic reactivity of the pollutant for the photoactivated surface. K obtained from the dark adsorption isotherms of the catalyst had been reported to be significantly different from the equivalent constant determined from kinetic data in photocatalysis (5, 8). Thus in practice, both constants are obtained from measured kinetic data. This expression is valid only in a batch reactor (25, 26). In the case of a continuous plug-flow reactor, the decrease of the concentration along the axial direction must be considered. If the mass transfer is negligible, there is no concentration gradient and the equation of continuity for the contaminant in cylindrical coordinates is written as follows (21, 22, 27):
ln Cout/Cin kKL 1 ) -K u Cin - Cout Cout - Cin
(7)
where the exponents “in” and “out” refer, respectively, to “inlet” and “outlet”, u is the superficial velocity (m‚s-1), L is the length of the annular reactor recovered by the catalyst (m), and dz is a differential element in the axial direction. Here, k is expressed in mmol‚m-3‚s-1. For each flow rate, a plot of (lnCout/Cin/(Cout - Cin)) versus ((L/u)1/(Cin - Cout)) should be linear and allow the two constants k and K to be determined (Table 2 and Figure 5). N-B: The correlation is done without considering the point of smaller concentration for a question of statistic weight. Indeed, the corresponding points, in the linear form, are quite far from the others on the graph: consequently, a small variation of this point can induce a big variation of the values of k and K. When k and K are determined, it is possible for a given inlet concentration to determine the outlet concentration using the solver Excel to solve eq 7. The model correlates quite well with the data (Figure 3). We note that, whatever the flow rate, the degradation rate seems to reach the same plateau at high inlet concentration. We also note that when the plateau is reached, the limiting degradation rate is only VOL. 41, NO. 8, 2007 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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dependent on the UV intensity in a given photoreactor, and therefore, the greater the UV intensity, the higher the limiting degradation rate (22). The values of k and K are dependent on the flow rate so they are not true kinetic constants. They intrinsically take into account the influence of the mass transfer step. Consequently, eq 7 cannot be considered as a true design equation. To determine the values of k and K, experiments at high flow rate with no mass transfer limitation must be carried out. This is not always technically possible. It can also lead to such a short residence time that the inlet and outlet concentrations may not be significantly different. This limits its use for scale-up. Consequently, it seems most appropriate to propose a new model where the mass transfer step is considered. Model II: The Mass Transfer Step is Considered. The internal mass transfer is assumed to be negligible. To confirm this assumption, the effectiveness (η) factors have been estimated for two extreme bulk concentrations (28). The Thiele moduli necessary for the calculation have been estimated for a Langmuir-Hinshelwood kinetic (14, 29). In both cases, the values of η were found to be rather close to one (higher than 0.99). The detailed calculation is presented as Supporting Information. Overall mass balances of the pollutant on the solid and the gaseous phases in a continuous plug-flow reactor lead to the following equations (19):
∂C Gaseous phase: u + kmav(C - Cs) ) 0 ∂z
(8)
KCs Solid phase: kmav(C - Cs) ) k 1 + KCs
(9)
Where km is the mass transfer coefficient in the radial direction (m‚s-1), av is the total effective catalyst area per unit volume of the reactor (m2‚m-3), and Cs represents the concentration of the pollutant near the surface of the catalyst (mmol‚m-3). These equations consider a radial concentration gradient between the surface and the bulk. A Langmuir-Hinshelwood mechanism is assumed. This system can be easily solved in the case of a first-order kinetic, i.e., KCs , 1 (19, 22, 30). Nevertheless, we have previously shown that, at high inlet concentration, this assumption is not valid. The mass transfer coefficients are estimated (Table 1) by the following semiempirical correlation specially developed for laminar flows in annular reactors (31):
Sh ) 1.029 × Sc0,33 × Re0.55 × (Ltot/deq)-0,472
(10)
Where, Sh is the Sherwood number, Sc is the Schmidt number, Re is the Reynolds number, Ltot is the length of the test zone, and deq is the equivalent diameter. Diffusivities in the air at 30 °C, which are necessary to calculate the Sherwood and Schmidt numbers, are estimated by the correlation of Fuller, Schettler, and Giddings (32) and were found to be 0.1017 cm2‚s-1 and 0.0897 cm2‚s-1, respectively, for PA and BA. The boundary values of the differential eq 8 are C(z ) 0)dCin and C(z ) L)dCοut. The resolution of this system is not analytically possible; nevertheless, an approached solution while using the software Maple V can be obtained thanks to series method. A development at the second (eq 11) and third (eq 12) order is given: 2912
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Cin - Cout ) Lkmav in 1 k C + + 2u K kmav
[
Cin - Cout ) Lkmav in 1 k C + + 2u K kmav
[
[
× 1+
x(
k 1 + K kmav
)
+
x(
1 k + K kmav
)
+
-Cin +
-Cin +
2
2
LK(kmav)3 4u
Cin +
x(
k 1 K k m av
x(
-Cin +
k 1 + K kmav
)
k 1 -C + + K kmav in
2
)
2
+
4Cin + K
4Cin K
]
4Cin K (11) 4Cin K
]
]
(12)
In these equations, the two unknown parameters are k and K. They are obtained by numeric resolution with the Solver Excel. For each experimental point (composed of Cin and Cοut), a target cell is defined as the difference between the experimental and the theoretical outlet concentrations. The theoretical outlet concentration is deduced from eqs 11 or 12. For each compound, two variable cells are used and refer to each constant. To facilitate the numeric resolution, the values obtained previously with Model I at the higher flow rate (where the mass transfer is less influent) are chosen as the initial values of the variable cells. The sum of the squares of the target cells is minimized by numeric resolution adjusting the variable cells. In this way, it is possible to find the best values of k and K for each compound. These values should be the true kinetic constants at given temperature, relative humidity, and UV intensity. The difference between the values of the constant at the second and the third orders is not significant (Table 3). The model correlates with very good agreement to the results (Figure 6). It seems interesting to compare the values of the kinetic constants between Model I and Model II. With Model I, K increases with the flow rate to the contrary of k. Normally, the values found for Model II should be the limits for Model I as flow rate increases. The model is a good prediction of the degradation kinetics whatever the flow rate. Similar plateaus as previously described were obtained with distribution of the data points on both sides of the curves. Moreover, it allows the determination of the limiting step. Determination of the Limiting Step. Rewriting eq 9:
kmav(C - Cs) )
kKCs ) k′Cs 1 + KCs
(13)
In the reactor, we assumed that k′ is constant in a differential element dz considering that Cs varies little. By rearranging eq 13:
Cs )
k m av C kmav + k′
(14)
and using the solver of Excel to solve this equation, it is possible to easily determine Cs for a given bulk concentration C in a differential element dz. According to Figure 7, the gradient concentration decreases when the bulk concentration increases because the mass transfer is more significant at low concentration where the active sites are not saturated.
FIGURE 6. Agreement between Model II and the experimental data for the PA (graph a) and the BA (graph b) at the various flow rates (2, 4, and 6 Nm3‚h-1). T ) 30 °C, RH ) 50%, I ) 40 W‚m-2. The curves correspond to the values predicted by Model II.
TABLE 3. Values of the Langmuir-Hinshelwood Constants for Model II. T ) 30 °C, RH ) 50%, I ) 40 W.m-2. propionic acid
butyric acid
order
kPA (mmol‚m-3‚s-1)
KPA (m3‚mmol-1)
R2 (%)
kBA (mmol‚m-3.s-1)
KBA (m3‚mmol-1)
R2 (%)
2 3
1.89 1.91
0.74 0.75
>99.99 >99.99
1.81 1.84
0.76 0.78
>99.99 >99.99
determine a maximal concentration near the surface of the ) below which the transfer is limiting. Accatalyst (Climit s cording to eq 14:
w Climit ) 2 Climit ) 2Climit s
FIGURE 7. Evolution of 1-Cs/C (in %) vs C for the butyric acid for various flow rates (2, 4, and 6 Nm3‚h-1): T ) 30 °C, RH ) 50%, I ) 40 W‚m-2. Moreover, when the flow rate increases, the gradient decreases as a consequence of a better mass transfer. By replacing Cs in the eq 8, we have the following:
∂C 1 ∂C u + ) -kobsC (15) C)0Su ∂z 1/k′ + 1/(kmav) ∂z Consequently, the observed degradation rate (kobs) is related to the kinetic rates (k′) and the mass transfer rates (kmav) by the following:
1 kobs
)
1 1 + k′ kmav
(16)
The transfer is the limiting step:
1 + KCs 1 k 1 1 1 w w Cs < < < (17) k′ kmav kK kmav kmav K Consequently, for any given flow rate, it is possible to
(
1 k kmav K
)
(18)
Thus, the maximal bulk concentrations below which the transfer is the limiting step are 7.9 mmol‚m-3 at 2 Nm3‚h-1; 4.6 mmol‚m-3 at 4 Nm3‚h-1 et 3.1 mmol‚m-3 at 6 Nm3‚h-1 for the PA and 8.5 mmol‚m-3 at 2 Nm3‚h-1; 5.0 mmol‚m-3 at 4 Nm3‚h-1 et 3.4 mmol.m-3 at 6 Nm3‚h-1 for the BA. Consequently, Model II allows the limiting step to be determined and then to know which elements of the reactor can be optimized (UV lamp, reactor geometry, diameter of the annulus, length of the photocatalytic zone, etc.). This model can be used very easily to achieve a scale-up of a treatment plant for a given effluent considering that the degradation rate constant k is proportional to the UV intensity or to its square root (17) at a medium humidity level.
Acknowledgments We are very grateful to M. Joseph Dussaud from the Ahlstro¨m Company who provided us with the catalytic support and to M. Ce´dric Vallet for all his help.
Nomenclature av
total effective catalyst area per unit volume of the reactor (m2‚m-3)
C
bulk concentration (mmol‚m-3)
Cs
concentration near the surface (mmol‚m-3)
deq
equivalent diameter ) do - di (m)
di
inner diameter of the annulus (m)
do
outer diameter of the annulus (m)
D
diffusivity in the air (m2‚s1)
I
intensity of the UV radiation (W‚m-2)
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degradation rate constant (mmol‚m-3‚s-1)
k
(m3‚mmol-1)
K
adsorption equilibrium constant
L
length of the reactor recovered by the catalyst (m)
Ltot
total length between the inlet and the outlet of the reactor (m)
r
degradation rate (M‚m-2‚s-1 or M‚m-3‚s-1)
Re
Reynolds number ) Fudeq/µ
RH
relative humidity (%)
Sc
Schmidt number ) µ/FD
Sh
Sherwood number ) kmdeq/D
T
temperature (°C)
u
superficial velocity (m‚s-1)
F
gaseous density (kg‚m-3)
µ
dynamic viscosity of the gas (Pa‚s)
φ
Thiele modulus
η
effectiveness factor
Supporting Information Available A detailed demonstration that the internal mass transfer is negligible. This material is available free of charge via the Internet at http://pubs.acs.org.
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Received for review October 4, 2006. Revised manuscript received January 23, 2007. Accepted February 7, 2007. ES062368N
