Photocatalytic Processes Using Solar Radiation. Modeling of

for a photocatalytic process which uses solar radiation in a batch or continuous slab reactor. ... H. I. Villafán-Vidales , S. A. Cuevas , C. A. ...
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Ind. Eng. Chem. Res. 1999, 38, 2940-2946

Photocatalytic Processes Using Solar Radiation. Modeling of Photodegradation of Contaminants in Polluted Waters Carlos A. Martı´n,‡ Giorgia Sgalari,†,§ and Francesco Santarelli*,† Dipartimento di Ingegneria Chimica, Mineraria e delle Tecnologie Ambientali, Universita` degli Studi di Bologna, Viale Risorgimento 2, 40136 Bologna, Italy, and Instituto de Desarrollo Tecnolo´ gico para la Industria Quı´mica (INTEC), Universidad Nacional del Litoral-CONICET, Guemes 3450 (3000) Santa Fe, Argentina

Presented here is an analysis of feasibility for a photocatalytic process which uses solar radiation in a batch or continuous slab reactor. The assumed geometry is representative of the simplest conceivable arrangement which can be used, and it is also the most suitable one to investigate the role of the relevant parameters in affecting radiative transfer and progress of the reaction. A procedure is developed for a preliminary investigation of the feasibility of these processes, and results are presented for situations typical of the photocatalytic degradation of toxic compounds in aqueous streams. Introduction Large attention is currently given to photocatalytic processes for their potential as effective environmental tools. Research efforts have been made toward a fundamental understanding of the interaction of the photons with the catalyst particles1-3 and, at the same time, specific reactions, mainly involving the treatment of waters with traces of highly toxic organic contaminants,4-6 have been under investigation. The body of literature in this latter area is very wide, starting from the pioneering works of Pruden and Ollis7 and Ollis et al.8 to the recent contribution by Theurich et al.,9 Martin et al.,10 and Cabrera et al.11 Despite the quantity and the quality of the results available in the literature, there are still several aspects in need of further analysis, if one wants to reach a full comprehension of the occurring phenomena and to develop reliable design procedures for practical applications. One of the main problems that still have to be solved is the photon source. As a matter of fact, the lack of a source that could operate effectively and cheaply is one of the main drawbacks for a wider diffusion of photocatalytic processes. Even if only a small fraction of its spectral distribution is effective in promoting a photocatalytic reaction, the sun satisfies the above-mentioned requirements. It is therefore worthwhile to explore the possibility of using its radiation in those situations where contaminated waters can be treated without any time constraint. The efforts made to develop water detoxification processes based on the use of solar energy have been recently reviewed by Bahneman,12 and it is apparent that the practical applications of these processes have to be considered within the reaction engineering framework. * To whom correspondence should be addressed. † Universita ` degli Studi di Bologna. ‡ Universidad Nacional del Litoral-CONICET. § Present address: Chemical Engineering Department, University of California Santa Barbara, Santa Barbara, CA 931065080.

An approach to the development of sounder tools for the analysis of a solar photocatalytic process is presented here, by considering a very simple arrangement that consists of a horizontal layer of liquidswhich can be quiescent or flowingsexposed to the solar incoming radiation through its upper face. This system allows us to get a reliable insight on the role of the various parameters in affecting the radiative transfer and the progress of the reaction. The analysis is based on the developmentsand the specialization for this class of processessof the approach presented by Stramigioli et al.13 for the general case of a photochemical direct reaction (absorption by the reacting species) in a quiescent fluid confined within a plane slab exposed to solar radiation. Basic Equations The photocatalytic degradation of a toxic contaminant present in a liquid stream is assumed to occur within a plane layer where the fluid is at rest or through which it is flowing, depending on the way the slab reactor is operated (batchwise or continuously). To model the system, one needs to consider the radiative transfer within a participating medium since the contaminated stream with the suspended catalyst particles interacts with the radiation through absorption and scattering. Assuming, as usual, that absorption is due only to the catalytic particles, no coupling occurs between the radiative transfer equation and the mass balance of the reacting species. In this case, in fact, radiative transfer depends only on the quantity and the distribution of the catalyst particles within the reactor, while it is independent of the progress of the reaction and the composition of the fluid phase. Radiative transfer and chemical reaction can therefore be examined sequentially. The fact that the progress of the reaction depends on the radiative-transfer equation via the local rate of radiant energy absorption (absorbed photons must be considered as immaterial reactants whose presence is required for the reaction to occur) is properly taken into account.

10.1021/ie980506e CCC: $18.00 © 1999 American Chemical Society Published on Web 06/25/1999

Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 2941

The mechanism of the whole reaction is very complex, and it has been recently reviewed by Alfano et al.14 This work provides an explanation for the different reaction orders, observed in experiments, and establishes a theoretical background for the design equations, thus realizing a substantial improvement with respect to the original work by Turchi and Ollis.5 (a) Radiative Transfer. Both the incoming radiation and the optical behavior of the irradiated fluid have to be characterized. It is well-known that the solar incoming radiation is affected by many variables such as the site location, the day of the year, the hour of the day, and the atmospheric conditions. The latter affect the incidence mode of the incoming radiation which can vary in the range defined by the extremes of collimated or beam radiation in clear days and diffuse radiation in cloudy days (see Figure 1). Examining these two extreme situations can be useful at this stage of the work, before modeling the actual conditions. Assuming that the dimensions of the exposed surface (length L and width W) are much larger than the depth (s) of the fluid layer, the system can be considered a plane slab. The analysis of the radiative transfer is reduced in this case to the solution of a one-dimensional problem. Following the approach proposed by Pasquali et al.1,3 and Alfano et al.,2 the heterogeneous medium through which the radiation travels can be treated as pseudohomogeneous with optical properties determined only by the catalyst particles. For any given effective wavelength λ and with omitting the index that refers to λ, the radiative-transfer equation (RTE) can therefore be written as

µ

∂I(τ,µ,φ,t) + I(τ,µ,φ,t) ) ∂τ ω0 2π 1 dφ′ -1I(τ,µ′,φ′,t)p(µ′,φ′fµ,φ) dµ′ (1) 4π 0

∫

∫

For a given incoming flux, the associated boundary conditions depend on the incidence mode considered and they can be written as

(i) beam radiation

Figure 1. Schematic of the investigated system.

It is worth noting that, even if the rate-of-change term does not appear in eq 1, the radiation field is dependent on time as the result of the time dependence of the boundary conditions. The radiation field within a participating medium is, in fact, always at steady state with respect to the system properties as well as to the boundary conditions;15 i.e., it follows, without inertia, any change occurring in the variables which affect it. The independent variable, τ, is the optical coordinate, a combination of the geometrical distance from the incidence surface and the optical properties of the system defined as

τ(z) )

∫0zβ(z′) dz′ ) β*∫0zccat(z′) dz′

where β ) (κ + σ) is the total spectral extinction coefficient of the medium [i.e., the sum of the absorption (κ) and the scattering (σ) coefficients]. It can be given as the corresponding specific (κ*, σ*) property times the concentration of the participating species (the catalyst particles here). The value τ0 of τ at the lower edge of the slab (z ) s) is known as the optical thickness and is a parameter that defines the magnitude of the slab’s interaction with the radiation. If the distribution of the catalyst concentration is uniform, τ and τ0 would result, respectively, in

τ(z) ) βz ) β*ccatz τ0 ) βs

I(0,µ,φ,t) ) I0 δ(µ - µ0(t)) δ(φ - φ0(t)) I(τ0,-µ,-φ,t) ) 0 (2) (ii) diffuse incident radiation I(0,µ,φ,t) )

I0 µ (t) π 0

I(τ0,-µ,-φ,t) ) 0 where I0 is the intensity associated with the incoming beam radiation (see Appendix A). The condition at the exposed surface (τ ) 0) is written in such a way that the incoming energy flux is the same for both modes. It is also assumed that reflection at the surface is negligible (i.e., all the incoming energy enters into the slab). The presence of a time-dependent term accounts for the apparent movement of the sun. The condition at the lower edge of the slab (τ ) τ0) simply accounts for a transparent, nonreflecting boundary.

(3)

(4)

Thus, the meaning of τ0 becomes apparent: as the ratio between the geometrical characteristic dimension of the system (s) and the mean free path of a photon (1/β), it is just the probability for a photon entering through the face z ) 0 not to escape through the opposite face z ) s. The value of τ0 can be evaluated for a given slab once the concentration of the catalyst particles of known optical properties is assigned. The local volumetric rate of radiant energy absorption (e˘ ′′′) is β(1 - ω0) times the incident radiation at that point, which accounts for the number of photons, traveling along any direction, which pass through the point. Once eq 1 is solved, the distribution of e˘ ′′′ at a given instant can be obtained as

∫02πdφ′∫-11I(τ,µ′,φ′,t) dµ′

e˘ ′′′(z,t) ) κ

(5)

With this quantity input into the kinetic equation of the photocatalytic reaction, the mass-balance equation of

2942 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999

the species to be mineralized becomes a closed equation, easily solvable. The procedure previously outlined actually refers to a single wavelength: it should be repeated for all the relevant wavelengths and the local rate of radiant energy absorption should result as the sum of the corresponding terms given by eq 5. A single wavelength will be considered in the following, thus assuming that only a very narrow band of wavelengths of the solar spectrum is effective for the reaction. (b) Slab Reactor. Since the slab reactor can be operated in a batch or a continuous mode, the two limiting cases will be considered in the following: perfectly mixed batch reactor (PMBR) and a continuous plug flow reactor (CPFR). It is worth remembering that the operation mode does not affect the solution of the radiative problem as long as the same distribution can be assumed for the catalyst’s particles concentration, (ccat). The local rate of disappearance of the species to be destroyed can be written (for reactions linearly dependent with the contaminant concentration) as

R(x,t) ) kc(x,t)[e˘ ′′′(x,t)]n

(6)

where c(x,t) is the instantaneous value of the local concentration of the species to be degraded, and the exponent n depends on the rate of holes generation on the surface of the catalyst particle [n ) 1 for the case of a low generation rate or n ) 0.5 in the high rate case, as shown by Turchi and Ollis5]. In both the investigated situations the changes of concentration over any cross section of the slab have been neglected, assuming perfect transversal mixing. Assuming also n to be constant over the slab depth, the former expression was integrated over the cross section to give the rate of generation per unit width of the slab:

R′(x,t) ) kc(x,t)

∫0s[e˘ ′′′(z,t)]n dz ) kc(x,t)F(t)

(7)

It must be noted that for n ) 1 F(t) is simply the rate of radiant energy absorption in the volume associated with a unit exposed surface and, therefore, it depends only on the value of τ0, whatever the distribution of ccat. When n * 1, on the contrary, this is not true any longer and F(t) has to be interpreted as a sort of radiation kinetic factor which depends on the distribution of e˘ ′′′ (and then on ccat, too). This is, in general, the case for the more complex situation, not considered here, where n varies with the position within the slab. The mass balance for the contaminant in the two investigated situations can be written as

{

PMBR dc ) -kc(t)F(t) dt c(0) ) c0

s

{

(

)

∂c(x,t) ∂c(x,t) +v ) -kc(x,t)F(t) ∂t ∂x c(x,0) ) cin c(0,t) ) c0

s

{

{

1 dγ ) -γF(η) Sr dη γ(0) ) 1

(9)

1 ∂γ 1 ∂γ + ) -γH(η) Sr ∂η Da ∂ξ γ(0,η) ) γe γ(ξ,0) ) γin

once the following dimensionless variables are defined:

γ)

e˘ ′′′ c t x ; η ) ; ξ ) ; a˘ ′′′ ) c0 TD L e˘ ′′′ 0

With regard to the Strouhal and the Damkohler numbers, Sr and Da, respectively, which appear in the dimensionless equations, their physical meaning is apparent from their definition, being n Sr ) TD[k(e˘ ′′′ 0 ) ] )

Da )

daylight length TD ) TR characteristic reaction time

mean residence time tr L n [k(e˘ ′′′ ) 0 ) ] ) v TR characteristic reaction time

with TD the length of the daylight time (see Appendix A). On the basis of the physical meaning associated with these two parameters, the following observations can be made: (1) Sr can be considered as an “external” process parameter since it includes the characteristic time TD of the boundary conditions. Everything else unchanged, its value depends on the latitude of the site and on the day of the year, reaching its minimum and maximum value at the winter and summer solstice, respectively. It is apparent that a solar process can be conceived only if Sr > 1 and that, furthermore, when Sr . 1, a continuous operation mode has to be preferred to a batch one to have a better exploitation of the solar radiation. (2) Da, on the contrary, can be considered an “internal” process parameter as its value is defined only by the mean residence time and by the reaction kinetics. Obviously, it appears only in the case of a continuous reactor and its value is dependent on the extent of the reaction that can be obtained: to have a good exploitation of the reactor volume, its value should therefore be larger than 1 but not too large. It is finally worth noting that in the case of a solar continuous photoreactor the ratio between the mean residence time and the daylight length is also important to assess the feasibility of a continuous operation. The ratio Da/Sr must be smaller than 1. To summarize, the significant dimensionless parameters for the investigated situation are the optical thickness τ0, for the radiative problem, and Sr and Da, for the linked reaction problem. Situation Investigated and Solution Procedures

(8) CPFR

or, in dimensionless form, as

The governing equations were solved for different values of the optical thickness τ0, of Sr and of Da (the latter only for the continuous reactor case). For the radiation problem, the optical thickness was varied from 0.4 to 4, thus investigating situations ranging from low levels (optically thin limit) to large

Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 2943

levels (optically thick limit) of participation of the irradiated medium. If reference is made to the optical properties of the catalyst particles as determined by Cabrera et al.,11 which are typical of Aldrich anatase TiO2, the values assumed for the optical thickness imply that, for a fluid depth s ) 1 cm, the particles concentrations vary in the range [10-5, 10-4] g/cm3. A value ω0 ) 0.87 was assumed for the single scattering albedo (as it is an intensive property of the material, it is independent of the quantity of catalyst). Scattering was assumed to be isotropic. This assumption may be questionable for this type of particle, because it is known that their actual behavior is more properly described in terms of diffuse reflection scattering.16 Nevertheless, isotropic scattering was considered for the sake of simplicity: at this stage of the analysis the influence of the phase functions is not very important.3 The relations used to evaluate the relevant parameters and the values used are given in Appendix A. With regard to the photocatalytic reaction, we considered the degradation of a model contaminant like chloroform. Its chemical equation is

CHCl3 + 1/2O2 + H2O f CO2 + 3HCl

Figure 2. The daily evolution of the power absorbed within the reactor for the case of beam radiation [F(t) for n ) 1].

(10)

The kinetics of this reaction has been studied by Martin et al.,10 who gave the following rate equation for the disappearance of chloroform:

(

R ) kcCHCl3(e˘ ′′′)0.5 with k ) 10.02

)

cm3 einstein‚s

0.5

(11)

The following values of Sr have then been evaluated, for the site considered, in three significant days of the year: Sr summer solstice equinox winter solstice

Figure 3. The daily evolution of the power absorbed within the reactor at the summer solstice in the case of beam and diffuse incoming radiation [F(t) for n ) 1].

22.80 17.74 12.16

Since Sr is always .1, it can be expected that a solar process can operate effectively through the entire year. For the solution of the radiative problem, the procedures given by Vestrucci et al.17 for direct radiation and by Spiga et al.18 for the diffuse one have been used. For all mentioned significant days and for all values of τ0 the solution was performed with a dimensionless time step ∆η ) 0.01. The mass balance of the species to be degraded was solved numerically according to an explicit solution scheme in the case of the batch reactor, while a solution based on the method of the characteristics was used for the continuous reactor (see Appendix B) Discussion of the Results (a) Radiative Transfer. The influence of different values of the optical thickness, of different incidence modes of the incoming radiation, and of operating in different days of the year has been explored. Figure 2 gives the daily evolution of the power absorbed within the slab for the unit exposed surface (i.e., F(t) for n ) 1) for different values of τ0 at the summer solstice and at the equinox, for the case of beam

radiation. Only half of the daylight length is reported for each of the investigated days, taking advantage of the symmetry with respect to noon (η ) 0.5) of the daily evolution of the solar irradiation. The effect of the variation of the incidence angle of the solar radiation during the day is apparent from the trend exhibited by all curves: they have their maximum just at noon. The role of the participation of the medium and that of the seasonal variation of the sun declination is also apparent. The energy absorbed within the slab is, for any day, larger the larger τ0 is and, for a given value of τ0, larger the closer the beam direction is to the vertical. Figure 3 gives the analogous curves evaluated at the summer solstice for the beam and the diffuse incidence modes, assuming, as previously said, that the same radiation flux is entering through the exposed surface. Similar trends result for both incidence modes: differences are quite low and they become smaller the larger τ0 is. Because the results for a larger variety of situations agree on this point, it can be safely assumed that differences between the two incidence modes are not important, especially in view of the larger uncertainties on other process variables. The same conclusions on the influence of the incidence mode on F(t) can be drawn also in the case n ) 0.5, which is relevant for the reaction studied in the following. Figure 4 gives the daily evolution of F(t) for n ) 0.5 and for two values of τ0 (0.4, 2) which are significant for an optically thin and an optically thick participating medium, respectively.

2944 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999

Figure 4. The daily evolution of the kinetic factor F for the case n ) 0.5.

Figure 5. The daily evolution of the dimensionless reactant concentration during selected days. Winter solstice, Sr ) 12.16; summer solstice, Sr ) 22.80; equinox, Sr ) 17.74.

It can therefore be concluded that once the incoming flux has been assigned, the incidence mode of the incoming radiation plays a minor role in assessing the performance of a solar photoreactor and that any of the two incidence modes can be used, at least in a preliminary analysis of the reactor. This conclusion has been drawn on the basis of the small differences between the solutions for the two incidence modes considered, but it appears even more appropriate when one remembers that the actual incidence of the solar radiation occurs as a combination of these limiting modes. Because of the linearity of the RTE, the relevant radiative quantities can be obtained, in general, as an appropriate combination of the solution for each of the two cases. Reference is made in the following only to the direct radiation incidence mode. (b) Photocatalytic Reaction. It is known19 that to have an effective exploitation of both the incoming radiation and the reaction volume, a value of τ0 larger than 1 but not too large is appropriate: the value τ0 ) 2 has been assumed in the subsequent analysis. The value of n has been kept constant to 0.5s consistently with the kinetic equation given in ref 10. With regard to this point, it must be noted that even if n can vary depending on the value of e˘ ′′′, n ) 0.5 is the less favorable condition in the expected range of variation of n for this type of reactor. Since no relationships are at the present known to account for the dependence, to assume a constant value n ) 0.5 is definitely a conservative assumption. The time evolution of the concentration of the contaminant in the batch reactor is given in Figure 5 for three selected days which are representative of the most and least favorable (the summer and winter solstice, respectively) and intermediate (equinox) insolation conditions.

Figure 6. Daily evolution of the reactant dimensionless concentration at the exit of the continuously operated slab reactor. Solid lines, winter solstice; dotted lines, summer solstice.

As could be expected from the values of Sr, a solar process is definitely conceivable because no matter which day of the year is considered, the reaction is almost completed just in the morning hours, thus making little or no use of most of the day’s radiation. It is worth noting that results are presented as a function of the dimensionless time: in such a way, the effectiveness of the operation can be evaluated for any specified day of the year. A comparison among different days, anyhow, is not possible in a straightforward way since the time scale TD changes with the day of the year. For this comparison, reference to the same exposure time would be more meaningful. The rescaling can easily be done, being η2 ) η1(TD,1/TD,z) and with the given law of change of TD along the year. Being Sr . 1, a continuous operation appears appropriate for a more effective use of all the incoming radiation. The analysis of the performance of a CPFR has been performed for two values (0.01 and 0.1) of Da/Sr. These values describe a continuous operation with rapid or slow transit through the reactor, respectively. The radiative conditions are the same considered in the case of the batch reactor. The daily evolution of the dimensionless exit concentration at the exit of the reactor is given in Figure 6 at the solstice days, for τ0 ) 2 and for the two values of Da/Sr. Results are reported for times larger than the residence time, i.e., after the initial content of the reactor has been washed out. It is evident that, for a given day, the conversion decreases with Da/Sr. Because the ratio between these two parameters can be interpreted as the ratio between residence time (tr) and the daylight length TD (a given value for a given day), a reduction of Da/Sr implies a reduction of the residence time and consequently of the conversion. To point out the effect of the seasonal variation when all the other process variables remain unchanged, comparisons were made between different days but for the same residence time. As TD depends on the day of the year, different values of Da/Sr must be properly chosen to account for the relevant changes in the value of Sr. The curve at the winter solstice for the same process conditions considered for Da/Sr ) 0.1 at the summer solstice is also given in Figure 6: the value Da/Sr ) 0.187 results from the reduction of the daylight length. The direct comparison of the results for the two cases

Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999 2945

clearly shows the effect of the reduction of incoming radiation flux at the winter solstice. Conclusions We presented a general procedure for the analysis of a photocatalytic process using solar radiation as the photon source for the treatment of an aqueous stream polluted with traces of toxic contaminants. The reactor volume investigated was horizontal plane slab, which is the simplest configuration to be used for solar processes even if it is known that more effective, but more expensive, arrangements can be used. The analysis of the general problem allowed us to identify the parameters relevant to the process. With regard to the radiative transfer, once a type of photocatalyst has been selected, the optical thickness τ0 is the most significant parameter because it gives a measure of the participation of the medium for the prescribed slab geometry and catalyst concentration. It was also shown that the radiative transfer within the slab can be treated satisfactorily, even under heavily simplifying assumptions. With regard to the reaction, the Strouhal and Damkohler numbers, ratios among the different time scales entering the process, allow one to assess the feasibility of the process and to identify the more effective operational mode. It must also be pointed out that, should more reliable kinetic data be available, additional parameters should be considered to account for the kinetics dependence on the rate of radiant energy absorption. To conclude, the present approach appears to be a useful tool in a preliminary phase of the analysis when the feasibility of the process has to be verified and the more convenient operation mode has to be selected. Its application to the photocatalytic degradation of a model contaminant like chloroform confirmed this property.

R ) reaction rate, defined by eq 7, mol cm-3 s-1 s ) thickness of the slab, cm n Sr ) TD[k(e˘ ′′′ 0 ) ], Strouhal number, dimensionless TD ) daylight time, s TR ) characteristic reaction time, s t ) time, s tr ) mean residence time in the continuous slab, s x ) axial coordinate along the slab, cm z ) distance from the face of the slab exposed to the solar radiation, cm Greek Letters R ) sun declination, deg β ) extinction coefficient, cm-1 ΦL ) latitude, deg γ ) c/c0, concentration, dimensionless η ) t/TD, time, dimensionless φ ) spherical coordinate, rad κ ) absorption coefficient, cm-1 λ ) wavelength, cm µ ) cosine of the angle between the direction of the radiation and the positive direction of the z axis µ0 ) cosine of the angle between the direction of the impinging beam and the positive direction of the z axis σ ) scattering coefficient, cm-1 τ ) optical coordinate, dimensionless τ0 ) optical thickness, dimensionless ω0 ) single-scattering albedo, dimensionless ω ) solar hour, deg ξ ) x/L, axial coordinate along the reactor, dimensionless

Appendix A Characterization of the Solar Irradiation Parameters. The zenith angle, θz, the angle between the beam from the sun and the vertical at the site, changes with the day of the year and the hour of the day and can be evaluated at any time using21

Acknowledgment

µ0(t) ) cos θz(t) ) sin R sin ΦL + cos R cos ΦL cos ω (A.1)

The support of the Commission of European Communities (Contract CI1-CT94-0035), MURST (Italy), Universidad Nacional del Litoral, and CONICET (Argentina) are gratefully acknowledged.

where ΦL is the latitude of the site, R is the sun declination (the day of the year), and ω the solar hour. The length of the sunlight hours can be evaluated through the following relationship,

Nomenclature a˘ ′′′ ) e˘ ′′′/e˘ ′′′ 0 , dimensionless local volumetric rate of radiant energy absorption c ) molar concentration, mol cm-3 ccat ) photocatalyst mass concentration, g cm-3 n Da ) L[k(e˘ ′′′ 0 ) ]/v, Damkohler number, dimensionless e˘ ′′′ ) local volumetric rate of radiant energy absorption, einstein cm-3 s-1 -3 e˘ ′′′ 0 ) characteristic value of e˘ ′′′ (for I ) I0), einstein cm s-1 e˘ ′′tot ) ∫s0[e˘ ′′′(z,t)] dz, total power absorbed per unit of exposed surface, (einstein cm-2 s-1) F ) ∫s0[e˘ ′′′(z,t)]n dz, radiation kinetic factor, (einstein cm-2 s-1)n H ) ∫10[a′′′]n d(z/s), dimensionless radiation kinetic factor I ) radiation intensity, einstein cm-2 s-1 I0 ) radiation intensity of the incoming beam solar radiation, einstein cm-2 s-1 k ) kinetic constant, (cm3 einstein-1 sn-1) L ) length of the slab, cm n ) reaction order with respect to e˘ ′′′, dimensionless p ) phase function, dimensionless

TD ) tsunset - tdawn ) 480 cos-1(-tgΦLtgR) (A.2) where cos-1 is expressed in degrees and TD results in seconds. The value ΦL ) 45° (a typical latitude of Central Italy) has been assumed in all the situations examined. Boundary Conditions for the RTE. In the case of direct radiation, the value I0 6.43 × 10-5 einstein/(m2 s), i.e., that of the spectral intensity at λ ) 350 nm, has been assumed for the intensity of the incoming beam.20 In the case of diffuse radiation, the value I0µ0(t)/π follows for the intensity of the radiation entering through the exposed face z ) 0 to match the incoming flux with the one of the previous case. Optical Properties of the Participating Fluid. For the TiO2 Aldrich anatase, Cabrera et al.11 derived experimentally the following values for the specific absorption and total extinction coefficients (350 nm):

k* ) 5.2 × 103 cm2 g-1 β* ) 4 × 104 cm2 g-1

2946 Ind. Eng. Chem. Res., Vol. 38, No. 8, 1999

which allows us to evaluate the single scattering albedo as ω0 ) (β* - k*)/β* ) 0.87, value which is independent of the particle concentration. The value of the total extinction coefficient, on the contrary, depends on the catalyst concentration and therefore different values of the concentration of the catalyst particles have been considered within the range used in kinetic studies: the following values resulted for the optical thickness τ0 ) β*ccats, for a value s ) 1 cm of the slab depth. ccat (g cm-3)

τ0

0.000 01 0.000 05 0.000 1

0.4 2.0 4.0

It has to be remembered that the relevant parameter for radiative transfer is τ0 and the aforesaid values of s and ccat have therefore to be considered only as indicative ones. It also has to be noted that a uniform distribution has been considered for the particle concentration, and therefore a linear relation exists between the geometrical coordinate z and the optical one τ. Appendix B For the case of the continuous operation of the slab, the plug flow model has been assumed and therefore the mass-balance equation for the degrading species, no matter if in dimensional or dimensionless form, can be solved using the method of the characteristics. The solution is given here for the dimensional form which allows a more direct readability of the results. Following the standard procedure which is typical of the cited solution method, it turns out that the distribution of the concentration of the reacting species is

∫0tF(t′) dt′]}u(x - vt) +

c(x,t) ) {cin(x - vt) exp[-k

t F(t′) dt′]} u(t - x/v) ∫t-x/v

{c0(t - x/v) exp[-k

(B.1)

where u(x) is the unit step function. As could be expected when t < (x/v), the concentration at any point depends on the concentration distribution existing upstream within the reactor at the start of the operation, while when t > (x/v), it depends on the composition of the reactor feed. In any case, these concentrations are modified according to their sun exposure stories which enter the integral in the exponential term. When specialized for x ) L, eq B-1 gives the concentration at the exit of the reactor and the ratio (x/v) is just the residence time tr. It must be warned that the exit concentration given in Figure 6 is only for t > tr, i.e., after the initial content of the reactor has been washed out. Literature Cited (1) Pasquali, M.; Santarelli, F. Proceedings of the 1st Conference on Chemical Process Engineering, AIDIC, Florence, Italy, 1993.

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Received for review August 3, 1998 Revised manuscript received March 15, 1999 Accepted February 17, 1999 IE980506E