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Quasi-isoactinic Reactor for Photocatalytic Kinetics Studies Alberto Brucato,* Franco Grisafi, Lucio Rizzuti, Antonino Sclafani, and Giuseppa Vella UniVersita` di Palermo, Dipartimento di Ingegneria Chimica dei Processi e dei Materiali, Viale delle Scienze, Ed.6 - 90128 Palermo, Italy
Photochemical reactors characterized by almost uniform values of the local volumetric rate of photon absorption (LVRPA), i.e., quasi-isoactinic photoreactors, are particularly suitable for assessing the influence of radiant field intensity in kinetic studies. In this work, Monte Carlo simulations have been performed to obtain LVRPA values in a flat photoreactor irradiated on both sides. This configuration appears to be particularly suitable for obtaining quasi-isoactinic conditions. The influence of catalyst albedo and scattering phase function is assessed, and the conditions for obtaining iso-actinicity are discussed. Finally, these conditions are related to an easy-to-measure parameter, namely, the photoreactor fractional transmittance. 1. Introduction
catalyst concentration. The first step is that of designing a photoreactor whose geometry is suitable for this purpose.
Photochemical and photocatalytic reactors are usually operated under conditions such that reagent and product concentrations, temperature, and especially radiation field distributions are not uniform.1-7 Through vigorous agitation and/or intense recirculation, a mixing degree could be reached that is high enough to make concentration and temperature inconsistencies negligible. However, there is no way to make the radiation intensity uniform by means of agitation, as photon paths are not affected by fluid motion. Moreover, photocatalytic reactions are often carried out under conditions of almost total absorption of the incident radiation, in order to obtain high values of the reaction rate. Under such conditions, local radiation intensity values typically vary by several orders of magnitude from point to point inside the photoreactor, making the kinetic interpretation of results troublesome. By acting on the geometry of the photon source/photoreactor system, it is possible to make all points inside the reactor be characterized by view factors of the photon source not too different from each other. Even then, however, it is not possible to remove the effects of radiation attenuation caused by the progressive absorption and scattering of the radiation by the chemical species and catalyst particles present. In particular, moving toward the inside of the reactor volume, away from the radiation inlet section, the radiation field intensity, and consequently the local volumetric rate of photon absorption (LVRPA), which is a very important parameter needed to link radiation distribution to local reaction kinetics,8 always decreases typically to extinction. Clearly, it would be convenient to perform kinetic investigations using photoreactors characterized by a radiant field intensity that is as uniform as possible. This goal could be reached by letting only a minimal fraction of incident radiation be absorbed in the reactor, for instance, by conveniently decreasing the volumetric concentration of the catalyst particles. However, under such conditions, reaction rates would be so small as to give rise to experimental difficulties. The aim of this work is that of setting up the conditions of an “iso-actinic” photoreactor, that is, a photoreactor in which the radiation field, and thus the LVRPA, can be considered uniform to a good extent, without dramatically decreasing the * To whom correspondence should be addressed. E-mail: abrucato@ dicpm.unipa.it. Tel.: +390916567216. Fax: +390916567280.
2. Quasi-Isoactinic Photorector The geometric arrangement proposed is akin to that described by Martin et al.9 for the case of a single-phase photochemical reactor, i.e., a simple plane slab irradiated from both sides, as depicted in Figure 1. In contrast to the Martin et al.9 arrangement, in the present case, the slab thickness is assumed to be very small in comparison to the other two dimensions in order to make side end effects small. Under such conditions, essentially a semi-infinite slab reactor is assumed. If the reactor were irradiated from one side only (e.g., only G′0 is present), the radiation intensity would decay with distance from that wall according to a law depending on the absorbing and scattering properties of the medium. This would, in turn, result in a decaying LVRPA that would attain its minimum at the nonirradiated reactor wall (curve LVRPA′ in Figure 1). Clearly, if the reactor were irradiated from the other side only, then for symmetry reasons, an identical but reversed LVRPA would have been obtained (curve LVRPA′′ in Figure 1). Through superposition of the effects, when both walls are irradiated, the LVRPA at each point inside the reactor is given by the sum of the two single irradiation values, resulting in curve LVRPAtotal of Figure 1, which shows much smaller variations than the two single irradiation curves. It might be worth remarking that, under single-side irradiation, significant uniformity of LVRPA can be obtained if radiation intensity is allowed to undergo only small variations from the front to the rear wall. This condition can be met if the medium has a small optical thickness, i.e., at very low catalyst loadings. As already remarked, in this case, the LVRPA would be quite uniform throughout the reactor but at the expense of lower values of photon absorption rate and, in turn, low reaction rates and consequent difficulties in the experimental assessment of reactor performance. In contrast, with double-side irradiation, a significant portion of the entering radiation can be usefully absorbed, thus achieving good reactor performance assessment yet maintaining good uniformity of LVRPA. The degree of uniformity attained for given conditions (reactor depth, catalyst albedo, particle size, and concentration) cannot be experimentally assessed, as this would require almost unfeasible local measurements of the LVRPA in the central part of the reactor volume. However, if a reliable radiation field model is available, such a model can be used for calculating
10.1021/ie0703991 CCC: $37.00 © 2007 American Chemical Society Published on Web 07/06/2007
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Vp ) RDp3
Ap ) βDp2
(3)
If xi-1 is the coordinate at which the last interaction occurred (where x is the distance from the reactor front plane), the depth of the new interaction is simply given by
xi ) xi-1 + li cos θi ) xi-1 - λ0 ln R1 cos θi
Figure 1. Geometric arrangement of the quasi-isoactinic photoreactor.
the LVRPA values at each point inside the reactor volume. For this purpose, a statistical approach, based on a Monte Carlo procedure, was employed in this work. 3. Monte Carlo Model The history of a statistically meaningful number of photons traveling inside a flat solid-fluid photoreactor was simulated under the hypothesis of uniformly distributed catalyst particles of the same size and a uniformly irradiated, indefinitely wide slab photoreactor of thickness L. Under the above assumptions, because of symmetry considerations, the only direction in which changes can take place is that of reactor depth. Moreover, in this simulation, the fluid phase is assumed to be perfectly transparent. Fundamentals of the Monte Carlo technique are well-known.10 For example, in the fields of photochemistry and photocatalysis, the Monte Carlo technique has been applied by Spadoni11 and Pasquali and Santarelli12 in investigations concerning photoassisted reactions. To describe the overall approach, let us follow the fate of a photon, starting from its first entrance into the photoreactor volume until it either escapes from the photoreactor or is absorbed by a photocatalyst particle. Assuming that the photon under consideration has just survived a scattering event inside the reactor, it is possible to calculate the length traveled by the photon before a further event (scattering or absorption) occurs. This length clearly depends on the probability of the photon colliding with a particle inside the photoreactor. This probability can be obtained by generating a random number, R1, so that, as the cumulative distribution of the probability density follows an exponential law, the length traveled by the photon between two subsequent interactions can be computed as
li ) -λ0 ln R1
(1)
where λ0 is the mean free path of the photons, which can easily be related to particle size and concentration as13
λ0 )
FpDp(R/β) m
(2)
where, in turn, m is the particle volumetric concentration, Fp is the particle density, Dp is the nominal particle diameter, and R and β are the volume and projected area shape factors defined by
(4)
where θi is the angle between the direction of photon travel and the x-axis direction. Clearly, a result of xi < 0 would mean that the photon had actually left the photoreactor from the front wall, and a result of xi > L would mean that the photon had escaped from the photoreactor rear wall. In all other cases, a new photon-particle interaction would occur at the specified location. If this is the case, another random number, R2, is generated and compared with the albedo, R. The photon is considered to be absorbed if R2 > R, whereas it is scattered if R2 < R. In the latter case, the scattering angle θi+1, defined by the latitude angle, φi+1, and the longitude angle, ωi+1, must be determined. To calculate the latitude angle, it is necessary to adopt a suitable phase function. There is no general consensus on the best phase function to adopt, as it depends on particle size and shape and surface properties. In this work, four different expressions for the phase function were employed: (i) pure backscattering (all scattered photons travel in the opposite direction than before interaction); (ii) pure forward scattering (all scattered photons travel in the same direction as before interaction); (iii) isotropic scattering (scattered photons have no preferential direction; hence, upon scattering, all information on their former direction is lost); and (iv) the well-known and theoretically sound “diffusely reflecting large sphere” phase function,14 described by the relationship
Φ(φ) )
8 (sin φ - φ cos φ) 3π
(5)
For the cases of pure backscattering and pure forward scattering, there was no need to generate a random number to assess the new direction, as this is deterministically defined by the model. In the case of isotropic scattering, the latitude angle φi+1 was obtained by generating the random number R3 (between 0 and 1) and solving eq 6 for φi+1.
R3 )
∫0φp(φ) dφ ) ∫0φ
sin(φ) 1 dφ ) (1 - cos φi+1) 2 2
(6)
In the case of the diffusely reflecting sphere phase function, the latitude angle φi+1 was obtained instead from eq 7
R3 )
∫0φp(φ) dφ ) ∫0φ
sin(φ) Φ(φ) dφ ) 2
1 (4φi+1 + 2φi+1 cos 2φi+1 - 3 sin 2φi+1) (7) 6π For both isotropic scattering and the diffusely reflecting sphere, because the same probability can be assigned to all directions,11 the longitude angle ωi+1 was obtained as
ωi+1 ) 2πR4
(8)
Knowledge of φi+1 and ωi+1 allows the calculation of θi+1, the angle of scattering relative to the x direction, by the geometric relationship
θi+1 ) cos φi+1 cos θi + sin φi+1 cos ωi+1 sin θi
(9)
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For each new photon, the procedure starts when the photon enters the reactor, i.e., x0 ) 0 and θ ) θ0, where θ0 is the angle between the direction of the incoming radiation and the x axis. In the present work, θ0 was always set to zero, i.e., uniform orthogonal irradiation was considered. The procedure ends when the photon either is absorbed or escapes from the photoreactor. To obtain the local volumetric rate of photon absorption (LVRPA), the photoreactor depth was suitably discretized into 100 slices of depth ∆x. The average (normalized) LVRPA in each slice was obtained by simply counting the number of absorption events that occurred in the slice and dividing this number by the incoming flux G0 and by ∆x/L. The number of photons ranged from 108 to 1010 and was always sufficient to obtain fully converged results. Even in the most demanding cases (large albedo values), CPU times never exceeded a few minutes on the common PC employed for the calculations. 4. Results and Discussion By using the above-described Monte Carlo simulation approach, LVRPA profiles were obtained for several values of catalyst albedo and reactor optical thickness. Results are reported as LVRPA normalized by LVRPA*, the mean value under the hypothesis of complete absorption inside the reactor, defined as
LVRPA* ) G0/L
(10)
As a consequence, a horizontal straight line at LVRPA/LVRPA* ) 1 would represent a situation in which only one side of the reactor was irradiated and all of the incoming photons were perfectly uniformly absorbed inside the reactor. Similarly, a horizontal straight line at LVRPA/LVRPA* ) 2 would characterize a condition in which the reactor was irradiated from both sides and all entering photons were uniformly absorbed. Consider now the number of photons absorbed, i.e., the reactive acts promoted, in a real reactor irradiated from one side only. In this case, the total photon absorption characterizing LVRPA/LVRPA* ) 1 would be achieved only if the optical density were large enough for radiation to be fully extinguished before reaching the other wall and the catalyst albedo were zero. In fact, in all other cases, some of the entering photons are bound to be lost from the system by transmission and/or reflection. This means that, in terms of the promoted reaction rate, an average normalized LVRPA value of 1 can be regarded as a good reference, as it represents the upper-bound (optimal) condition for a normally irradiated photoreactor. Results of the simulations performed are reported in Figure 2a-c as LVRPA/LVRPA* versus x/L for three values of R and five values of L/λ0, with the well-known diffusely reflecting sphere used as the scattering phase function. In particular, Figure 2a reports the simulations performed for R ) 0.1, a realistic condition if the photocatalyst is TiO2 and the irradiation wavelength is lower than 350 nm15. In this case, for L/λ0 ) 0.5 (solid line), a straight line is obtained at a value of at LVRPA/LVRPA* ) 0.74, not much lower than 1. For L/λ0 ) 1, an almost flat horizontal profile is obtained, at a value between 1 and 2, showing better total photon absorption than the ideal extreme for irradiation on one side only. In other words, the required LVRPA uniformity condition is fully attained, with the reaction rate still maintained as large as that in an optically dense photoreactor with single irradiation. For L/λ0 ) 2 (short dashed line) the curve is not flat, but a fair iso-actinicity is still shown, as the percentage deviation of
Figure 2. Simulated values of LVRPA versus reactor thickness at various values of catalyst albedo and optical thickness.
the minimum from the maximum is 33%, with a mean value of 1.67. For higher values of L/λ0, the reactor can no longer be considered iso-actinic, as can be noted by observing the relevant curves, although the LVRPA variation is nevertheless smaller than the several-orders-of-magnitude variation often found in photocatalytic reactors employed for kinetic studies. The curves obtained for R ) 0.5 and for the same phase function are reported in Figure 2b. This is again a realistic condition, as it corresponds to the albedo exhibited by TiO2 for photon wavelengths of about 370 nm.15 The increase of R gives
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Figure 3. Standard deviation of LVPRA versus optical thickness for various values of catalyst albedo and for the four phase functions used for simulations.
rise to lower values of LVRPA and better uniformity. For L/λ0 ) 2, the percentage deviation of the minimum from the maximum decreases to 20.4%, with a mean value of 1.34. For L/λ0 ) 4, the curve starts to exhibit two local maxima in the LVRPA close to reactor walls. This feature was previously obtained theoretically4,16-19 and observed experimentally.4 Again, for higher values of R, the reactor does not behave in an iso-actinic manner. The values obtained for R ) 0.9 and the same phase function are reported in Figure 2c. Still lower values of LVRPA are observed, as expected. However, a singular feature occurs, as the above-described relative maxima are more pronounced, as can be seen for L/λ0 ) 8. Moreover, as L/λ0 decreases, these maxima move away from reactor walls, first mitigating LVRPA variations, as can be observed for L/λ0 ) 4, and then generating LVRPA values that are larger in the reactor midplane than near the reactor walls, as can be observed for L/λ0 e 2. This feature leads, in turn, to extended LVRPA uniformity for values of L/λ0 up to 4. As LVRPA uniformity is the major concern of this work, its quantification is needed for discussion of the results. The choice adopted here is that of employing the standard deviation of the LVRPA distribution, defined as
s)
x
∑(LVRPAi - LVRPA)2 n-1
(11)
and applied to the 100 discrete values obtained. The calculation was performed for all simulations. Henceforth, we consider as quasi-isoactinic all reactors having a standard deviation smaller than 0.1. Figure 3a presents a plot of s versus L/λ0 for the diffuse reflecting sphere phase function and for values of R ranging from 0 to 0.9. Examination of this figure leads to the following conclusions: The worst case is R ) 0, which corresponds to the already-mentioned zero reflectance model.13 Even in this disadvantageous case, for L/λ0 < 1.3, the reactor is quasiisoactinic. This is a very good result because it could be considered as the lower limit. For the more realistic case of R ) 0.5, in fact, L/λ0 has a value of 1.9. It can be observed also that all curves for R values higher than 0.5 exhibit an S-shaped trend. This is obviously due to the already-reported presence of the local maxima values of LVRPA near the reactor walls. In turn, this feature contributes to the increase of the maximum L/λ0 value for which quasi-isoactinic conditions are obtained. Similar behavior can be observed in Figure 3b-d, where the data for the three other phase functions investigated here (isotropic, pure backscattering, and pure forward scattering) are plotted. Of course, in all cases, the curves for R ) 0 coincide, as no scattering occurs for this extreme condition and the results are independent of the phase function employed. It is worth noting that, in all cases, the R ) 0 condition is the worst in terms of LVRPA uniformity. In other words, as might have been expected, light scattering always has a redistributing effect that
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Figure 4. Influence of the phase function on the standard deviation for four values of the catalyst albedo.
improves LVRPA uniformity, although at the expense of the total photon absorption rate, independently of the phase function. Another way of observing the influence of the phase function is shown in Figure 4a-f, where the standard deviation is plotted against L/λ0 for six values of R and for the four phase functions employed in the simulations. One can see that, for R ) 0.1 (Figure 4b), the four curves still practically coincide. Also, for R ) 0.25 (Figure 4c), no significant differences appear. Slight differences can be observed for R ) 0.5 (Figure 4d), with the appearance of the S-shaped trend. More significant differences can be noted for higher values of R (Figure 4e,f), although the values of L/λ0 appear to be fairly close, but for the forward phase function. In conclusion, it appears that, throughout the entire range from R ) 0 to R ) 0.9, the influence of the type of phase function used is not very important. Only for the
extreme cases of the forward phase function and high values of R (i.e., R > 0.5) can the differences be considered significant. To provide a useful operating tool, the same results of Figure 4 are plotted in Figure 5a-d as the standard deviation, obtained as defined by eq 11, versus the fraction of light transmitted through the reactor when only one side is irradiated, T ) GL/ G0, where GL and G0 are the photon fluxes at x ) L and at x ) 0, respectively. This last parameter is, in fact, a quantity that can easily be accessed experimentally. Indeed, it can be measured by placing a radiation detector on the nonirradiated side of the reactor and measuring the transmitted photons at several catalyst concentrations. The ratio of the measured signal to that obtained by the same detector in the absence of catalyst particles provides a good approximation of the fractional transmittance T, i.e., the abscissa of Figure 5. With this
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Figure 5. Photon transmittance for single-side irradiation versus LVRPA standard deviation for four catalyst albedo values and the four phase functions used.
information in hand, if an estimate of the catalyst albedo to the radiation of interest is available and a reasonable assumption on the phase function shape can be made, one can use Figure 5 to immediately estimate the uniformity level attained in the reactor when both sides are irradiated. It is worth noting that the case of no scattering (R ) 0) systematically gives rise to the poorest LVRPA uniformity. In other words, an increase of R always results in an improvement of iso-actinicity, independently of the particular scattering phase function exhibited by the catalyst particles. For this limiting case, the prediction is extremely simple, as the value of LVRPA/ LVRPA* and its mean can be computed on the basis of the very simple zero reflectance model13
LVRPA ) -ln[T x/L + T (1-x/L)] LVRPA*
(12)
LVRPA 〈LVRPA* 〉 ) 2(1 - T)
(13)
and the relevant iso-actinicity level can be immediately assessed by simple hand calculation. This clearly results in an underestimation of the actual iso-actinicity level, as, in the likely case of polychromatic irradiation, there will be radiation wavelengths able to promote the photocatalytic reactions and characterized by all albedo values from almost 0 to almost 1. On the basis of the results in Figure 5, all of the radiation with a significant albedo will contribute to better LVRPA uniformity than predicted by the limiting case of R ) 0.
It is finally worth noting in Figure 5 that, if one loads the reactor with catalyst particles so that about 30% of the singleside irradiation is transmitted, surely, the LVRPA standard deviation will be less than 10% or, conversely, an average uniformity better than 90% will be obtained with the doublesided irradiation. The catalyst concentration needed for a given photoreactor thickness strongly depends on the catalyst particle size (eq 2). The primary TiO2 particles are typically very small, but when suspended in water, they tend to form aggregates. Therefore, the value of Dp that should be used in eq 2 is that of the aggregates. This value depends on a dynamic balance between aggregation and aggregate-breakage kinetics and, hence, on complex phenomena involving turbulent stresses and particle surface properties, clearly resulting in a difficult-to-predict particle size distribution. It is for this reason that Figure 5 was provided, as it is much easier to irradiate only one side of the photoreactor and to measure the amount of light transmitted at various catalyst loadings than to estimate all parameters needed for a fully predictive application of the information reported in Figure 4. However, if an order-of-magnitude estimate is sought, then one can consider that typical average aggregate sizes in TiO2 suspensions have been found13 to be in the range of 0.5-1.0 µm. To let 30% of incident light be transmitted, an L/λ0 value of 1.2 is required for R ) 0 (Brucato and Rizzuti,13 eq 16), and for a reactor thickness of L ) 0.01 m, one obtains λ0 ) 0.0083 m. Finally, for R ) 0, assuming an average catalyst size of
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0.75 µm, Fp ) 3000 kg/m3, R/β ) 0.667, then from eq 2, a catalyst concentration of 0.18 kg/m3 (0.18 g/L) is required. Clearly, a 5-cm-thick reactor would require a catalyst concentration exactly 5 times smaller.
Acknowledgment This research was supported by a grant to INCA “Consorzio Interuniversitario La Chimica per l’Ambiente” on Legge 488 funds, Research Project 3: “Rimozione di inquinanti mediante processi a membrana e fotocatalitici”.
5. Conclusions
Literature Cited
A simple arrangement for obtaining quasi-isoactinic conditions is proposed. It consists of a simple slab photoreactor irradiated on both sides. Monte Carlo simulations of the system were performed. Results show that quasi-isoactinic conditions can be obtained provided that the photocatalyst loading is reasonably low. Notably, double irradiation allows iso-actinicity to be obtained with total radiation absorption levels (i.e., amount of photoreactions promoted) on the same order as, or even larger than, those obtained with single-side irradiation and large catalyst loadings. The conditions for iso-actinicity are discussed and finally related to a simply measured parameter as the reactor fractional transmittance.
(1) Pozzo, R. L.; Brandi, R. J.; Giombi, J. L.; Cassano, A. E.; Baltana´s, M. A. Fluidized bed photoreactors using composites of titania CVD-coated onto quartz sand as photocatalyst: Assessment of photochemical efficiency. Chem. Eng. J. 2006, 118, 153. (2) Yang, Q.; Ang, P. L.; Ray, M. B.; Pehkonen, S. O. Light distribution field in catalyst suspensions within an annular photoreactor. Chem. Eng. Sci. 2005, 60, 5255. (3) Li, Puma, G. Modeling of thin-film slurry photocatalytic reactors affected by radiation scattering. EnViron. Sci. Technol. 2003, 37, 5783. (4) Pareek, V.; Brungs, M. P.; Adesina, A. A. A New Simplified Model for Light Scattering in Photocatalytic Reactors. Ind. Eng. Chem. Res. 2003, 42, 26. (5) Alfano, O. M.; Bahnemann, D.; Cassano, A. E.; Dillert, R.; Goslich, R. Photocatalysis in water environments using artificial and solar light. Catal. Today 2000, 58, 199. (6) Cabrera, M. I.; Alfano, O. M.; Cassano, A. E. Scattering effects produced by inert particles in photochemical reactors. 2. A parametric study. Ind. Eng. Chem. Res. 1995, 34, 500. (7) Alfano, O. M.; Cabrera, M. I.; Cassano, A. E. Modeling of light scattering in photochemical reactors. Chem. Eng. Sci. 1994, 49, 5327. (8) Brandi, R. J.; Alfano, O. M.; Cassano, A. E. Evaluation of Radiation Absorption in Slurry Photocatalytic Reactors. Part. I. Assessment of Methods in Use and New Proposal. EnViron. Sci. Technol. 2000, 34 (12), 2623. (9) Martin, C. A.; Alfano, O. M.; Cassano, A. E. Decolorization of water for domestic supply employing UV radiation and hydrogen peroxide. Catal. Today 2000, 60, 119. (10) Hammersley, J. M.; Handscomb, D. C. Monte Carlo Methods; Chapman and Hall Ltd.: London, 1983. (11) Spadoni, G.; Bandini, E.; Santarelli, F. Scattering Effects in Photosensitized Reactions. Chem. Eng. Sci. 1978, 33, 517. (12) Pasquali, M.; Santarelli, F. Radiative Transfer in Photocatalytic Systems. AIChE J. 1996, 42 (2), 532. (13) Brucato, A.; Rizzuti, L. Simplified Modelling of Radiant Fields in Heterogeneous Photoreactors. Part I - Case of Zero Reflectance. Ind. Eng. Chem. Res. 1997, 36, 4740. (14) Siegel, R.; Howell, J. R. Thermal Radiation Heat Transfer; McGraw-Hill Book Company: London, 1972. (15) Brucato, A.; Grisafi, F. Economic evaluation of UV sources for photocatalytic applications. J. AdV. Oxid. Technol. 1999, 4 (1),47. (16) Brucato, A.; Cassano, A. E.; Grisafi, F.; Montante, G.; Rizzuti, L.; Vella, G. Estimating Radiant Fields in Flat Heterogeneous Photoreactors by the Six-Flux Model. AIChE J. 2006, 52, 3882. (17) Brandi, R. J.; Alfano, O. M.; Cassano, A. E. Rigorous model and experimental verification of the radiation field in a flat-plate solar collector simulator employed for photocatalytic reactions. Chem. Eng. Sci. 1999, 54, 2817. (18) Brandi, R. J.; Alfano, O. M.; Cassano, A. E. Modeling of radiation absorption in a flat plate photocatalytic reactor. Chem. Eng. Sci. 1996, 51, 3169. (19) Romero, R. L.; Alfano, O. M.; Cassano, A. E. Cylindrical photocatalytic reactors. Radiation absorption and scattering effects produced by suspended fine particles in an annular space. Ind. Eng. Chem. Res. 1997, 36, 3094.
Notation Ap ) particle area (m2) ap ) projected area of one catalyst particle (m2) Dp ) catalyst particle diameter (m) G ) radiant flux (einsteins-1 m-2) l ) length traveled by a photon (m) L ) photoreactor thickness in the radiant flux direction (m) LVRPA ) local volumetric rate of photon absorption (einsteins-1 m-3) LVRPA ) mean value of the LVRPA (einsteins-1 m-3) m ) catalyst concentration (kg m-3) n ) number of intervals of reactor discretization R ) catalyst albedo, the ratio between the number of scattered photons and the total number of photons (scattered + absorbed) interacting with the catalyst surface R1, R2, R3, R4 ) random numbers in the range of 0-1 s ) standard deviation T ) fraction of light transmitted when only one side is irradiated Vp ) particle volume (m3) x ) Cartesian coordinate Greek Symbols R ) shape factor for particle volume (particle volume ) Rdp3) β ) shape factor for particle projected area (particle projected area ) βdp2) φ ) latitude angle (rad) λ0 ) length constant (eq 1) (m) θ ) scattering angle (rad) Fp ) catalyst density (kg m-3) ω ) latitude angle (rad)
ReceiVed for reView March 16, 2007 ReVised manuscript receiVed May 10, 2007 Accepted May 17, 2007 IE0703991
