Radiation Absorption and Optimization of Solar ... - ACS Publications

Environ. Sci. Technol. 2010, 44, 5112–5120

Radiation Absorption and Optimization of Solar Photocatalytic Reactors for Environmental Applications ´ RQUEZ,† JOSE COLINA-MA F I D E R M A N M A C H U C A - M A R T Í N E Z , ‡ A N D G I A N L U C A L I P U M A * ,§ Chemical Engineering Department, Universidad de Cartagena, Cartagena, Colombia, Chemical Engineering School, Universidad del Valle, GAOX Group, Cali, Colombia, and Photocatalysis & Photoreaction Engineering, Department of Chemical and Environmental Engineering, The University of Nottingham, Nottingham NG7 2RD, United Kingdom

Received January 13, 2010. Revised manuscript received May 20, 2010. Accepted May 25, 2010.

This study provides a systematic and quantitative approach to the analysis and optimization of solar photocatalytic reactors utilized in environmental applications such as pollutant remediation and conversion of biomass (waste) to hydrogen. Ray tracing technique was coupled with the six-flux absorption scattering model (SFM) to analyze the complex radiation field in solar compound parabolic collectors (CPC) and tubular photoreactors. The absorption of solar radiation represented by the spatial distribution of the local volumetric rate of photon absorption (LVRPA) depends strongly on catalyst loading and geometry. The total radiation absorbed in the reactors, the volumetric rate of absorption (VRPA), was analyzed as a function of the optical properties (scattering albedo) of the photocatalyst. The VRPA reached maxima at specific catalyst concentrations in close agreement with literature experimental studies. The CPC has on average 70% higher photon absorption efficiency than a tubular reactor and requires 39% less catalyst to operate under optimum conditions. The “apparent optical thickness” is proposed as a new dimensionless parameter for optimization of CPC and tubular reactors. It removes the dependence of the optimum catalyst concentration on tube diameter and photocatalyst scattering albedo. For titanium dioxide (TiO2) Degussa P25, maximum photon absorption occurs at apparent optical thicknesses of 7.78 for CPC and 12.97 for tubular reactors.

Introduction Heterogeneous photocatalysis based on TiO2 and modified semiconductor photocatalysts has received a great deal of attention in the literature as an environmentally friendly method for the treatment and purification of lightly contaminated water and air (1, 2), self-cleaning surfaces (2, 3), and as a route for sustainable energy production (4, 5). Irradiation of TiO2 with radiation of energy higher than the * Corresponding author phone: +44 (0)115 951 4170; fax: +44 (0)115 951 4115; e-mail: [email protected]. † Universidad de Cartagena. ‡ Universidad del Valle. § The University of Nottingham. 5112

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band gap of the semiconductor (e.g., λ < 384 nm for TiO2 anatase) produces electron-hole charges on the semiconductor which can initiate reduction and oxidation reactions. Oxidation of TiO2 adsorbed water, by surface holes, produces hydroxyl and other radical species which are responsible for the titania’s wide-ranging activity toward large classes of contaminants (e.g., aromatics, halogenated hydrocarbons, pesticides, endocrine disrupting chemicals, inorganics, and others) (1-3, 6) and for the inactivation of microorganisms and toxins (e.g., coliforms, viruses, microcystins) (2, 3, 7, 8). Reduction of water by the photogenerated electrons and simultaneous oxidation of biomass (waste) and/or organic pollutants, in the absence of oxygen, has been proposed as a method for hydrogen production from renewable sources (9). The photogenerated electrons can also reduce harmful heavy metals present in the environment (1). The expansion and the technological application of heterogeneous photocatalysis in the above environmentally relevant processes require the development of efficient photoreactors. Since sunlight is the only free source of photons, solar photoreactors are desirable. Photons are necessary reactants or initiators; therefore, the knowledge of the radiation field in a photoreactor is crucial to maximize their efficient use, products yield, and selectivity (10). However, due to lack of widely accessible engineering knowledge current photoreactors for environmental applications and sustainable energy production are designed by experience rather than from rigorous scientific principles. Furthermore, intrinsic kinetic parameters of photocatalytic reactions can be determined only with the knowledge of the radiation field in the photoreactor (11-14). In general, optimal reactor geometry, photon distribution, and catalyst loading are interrelated concepts which require accurate modeling of the radiation field, in the photoreactor. The spatial distribution of photons, i.e., the local volumetric rate of absorption (LVRPA) depends on the photon source, the optical properties of the system, the distribution of the catalyst and the reactor geometry. Several approaches have been proposed to calculate the LVRPA in photocatalytic reaction systems in which absorption and scattering of photons occur. The more rigorous studies have developed a numerical solution of the radiative transfer equation (RTE) (10, 11). This procedure implies a thorough mathematical work for describing the radiant field for each given reactor geometry and the emission model of the photon source. Other studies have used a semiempirical approach (15) that calculates the LVRPA by fitting model parameters to experimental data, in a attempt to avoid solving the RTE. Some authors (16) proposed a model without adjustable parameters considering all scattering directions to describe the photodegradation of the herbicide carbaryl. Others (17) modeled the radiation field in a parabolic trough solar photocatalytic reactor. The P1 approach was proposed for solving the RTE in solar tubular photoreactors (18). Other studies have considered Monte Carlo and the finite volume methods (19, 20) to solve the RTE. For practical purposes in photocatalytic processes, it is necessary to use simple models that describe the radiation field and photon absorption in most reactor geometries and that can determine the essential parameters needed for design and optimization. The Six-Flux absorption-scattering model (SFM) has been proposed to estimate the LVRPA in heterogeneous photocatalytic systems offering simplicity and accuracy. This model is based on establishing probabilities for six different Cartesian scattering directions. The SFM was applied to estimate the LVRPA in a flat plate (21), in annular 10.1021/es100130h

 2010 American Chemical Society

Published on Web 06/09/2010

reactor considering both the direct and diffuse components of solar radiation has been presented in our previous work (22). Furthermore, it was assumed that the photon rays were parallel, and they were vertically directed toward the photoreactor wall, to simulate the solar irradiation conditions at noon when the photoreactors receive the largest amount of energy. In the presence of a sun tracking device the perpendicularity of the rays over the reactor wall and reflector surface can be ensured for most of the day. The tubular reactor receives radiation from the upper hemicylindrical wall only, exposed to the sun, whereas the CPC also receives the radiation reflected by the two involute reflectors (Figure 1). Since tubular and CPC reactors are long tubes, the radiant field was modeled in a 2-D space assuming no variation of the incident radiation along the axial direction (z-axis). The coordinates’ origin was located at the center of the transversal circular section of the tube, (Figure 1). Ray-tracing was used to model the reflected radiation by the CPC involutes. The collectors geometry was described by (26) xCPC ) (RR(sin t - t cos t)

(1)

yCPC ) -RR(cos t + t sin t)

(2)

where t is the angle measured from the negative y-axis (Figure 1b). From these the position vector r of each point on the CPC is r ) xi + yj

(3)

The tangent and the normal unitary vectors were calculated as follows

FIGURE 1. Geometry of (a) tubular and (b) CPC solar photoreactors and direction of travel of direct and reflected solar radiation. reactors to model the photocatalytic degradation of phenylurea and triazine herbicides (12, 13), and in conjunction with ray tracing to model the photocatalytic mineralization of commercial herbicides (used in sugar cane crops) in a pilot-scale, solar, compound parabolic collector (CPC) reactor (22). This geometry (Figures 1 and S1, Supporting Information (SI)) is one of the most efficient for solar photocatalytic applications (23) and has been used extensively in many pilot and full scale studies. Some excellent reviews (10, 24, 25) have reported on engineering aspects of solar and CPC photocatalytic reactors. However, to date, there are no fundamental studies that can be used for the description of the radiation field (LVRPA) in CPC photocatalytic reactors and for the design and optimization of these reactors. In this study, ray tracing was coupled with the SFM to model the complex radiation field in solar CPC and tubular reactors. The model, validated with data from literature, provides a simple and systematic approach for the design and optimization of solar photocatalytic reactors for environmental applications. Modeling of Radiation Field in Solar Tubular and CPC Reactors. Incident Radiation Flux. For the purpose of this work and for simplification purposes the reactors were considered to be irradiated by the direct component of solar radiation. The diffuse component of solar radiation was neglected to avoid unnecessary complications due the random nature of its propagation through space. Therefore, the model presented here is more suited to zenith angles smaller than 45° in which the direct component of solar radiation prevails. An analysis of the radiation field in a CPC

T)

dr/dt |dr/dt|

(4)

N)

dT/dt |dT/dt|

(5)

Combining eqs 3-5, the angle β of the normal vector measured from the positive x-axis, at any point of the CPC surface, was calculated T ) ((sin t)i + cos tj

(6)

N ) ((cos t)i - sin tj

(7)

Ny ( |N| ) ) arcsin(-sin t) ) {π t- t x x< g0 0

β ) arcsin

(8)

The incidence and reflection angles were estimated from the angle of the axis normal to the CPC surface (Figure 1b) assuming specular ray reflection, eqs 9-11 Ri ) 2βi + θ1 - π/2

(9)

mi ) tan Ri

(10)

yi ) mi(x - xi-1) + yi-1

(11)

where i is the number of consecutive reflection of a ray on the CPC surface, θ1 is the angle of the incident ray from the y-axis, and R is the reflected ray angle from the positive x-axis. After each reflection event an iterative process was used to establish if the reflected ray intersected the reactor wall or the CPC surface again. The equation of the reactor wall circumference is xR2 + yR2 ) RR2 VOL. 44, NO. 13, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

(12)

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Equation 11 was substituted in eq 12 to yield a second order equation that was solved with eqs 13-17 ai ) 1 + mi2

(13)

bi ) 2mi(yi-1 - mixi-1)

(14)

ci ) (yi - mixi-1)2 - RR2

(15)

δ ) 2√RR2 - xp2

(23) (24)

rp ) δ/2 - r sin θ

yW ) yi-1 + mi(xW - xi-1)

(17)

In addition, the radiation reflected by the reflectors (CPC reactor only) contributed to the LVRPA. The distance δ in the reactor tube crossed by each reflected ray was (Figure 1b) δ ) √(xi+1 - xi)2 + (yi+1 - yi)2

(25)

yCPC,estimated ) yi-1 + m1(xCPC - xi-1)

(18)

The rectangular coordinates of the points where a reflected ray intercepted the reactor wall were estimated through the ray-tracing eqs 9-17. The angular coordinates of the points of the reactor wall intercepted by the reflected ray, θk,0 and θk,F (Figure 1b), were

fCPC,error ) |yCPC - yCPC,estimated |

(19)

θk,0 ) arctan(yi /xi)

(26)

θk,F ) arctan(yi+1 /xi+1)

(27)

I0,reflected ) I0ψn

(20)

where I0 is the solar incident photon flux, ψ is the material reflectivity, and n is the number of consecutive reflections of the ray on the involute surface. Estimation of the LVRPA. The LVRPA was estimated with the SFM (13, 21, 22) which also include the underlying assumptions and the definition of the model parameters I0 -1+ [(ω λωcorrωcorr(1 - γ) corr

2 )e-r /λ √1 - ωcorr p

ωcorr

2 + γ(ωcorr - 1 - √1 - ωcorr )erp/λωcorr] (21)

I0 is the incident solar radiation flux reaching the reactor walls, either directly or reflected by the collector. The corrected scattering albedo ωcorr was calculated from the optical properties of the TiO2 suspension (SI), whereas rp is the coordinate distance in the direction of traveling of the incident ray (Figure 1). The extinction length, λωcorr, and the parameter γ depend on the apparent optical thickness, τapp, which in turn depends on the photocatalyst concentration, ccat, the extinction coefficient, and the reactor thickness δ traveled by the ray (SI). The estimation of the LVRPA at a given position of polar coordinates (r,θ) required different calculation procedures for the direct incident radiation (CPC and tubular reactor) and for the reflected radiation (CPC reactor only). The change of direction due to the ray refraction at the photoreactor walls was neglected for simplification purposes. 9

(22)

(16)

with xCPC and yCPC calculated with eqs 1-2, and yCPC,estimated calculated with the reflected ray equation, eq 17. The reflection point was the one showing the minimal error over the surface of the involute. After performing the necessary iterations, the coordinates where the ray was reflected by the CPC surface as well as where it intersected the reactor wall were calculated. With these known, the distance δ traveled by the ray within the reactor tube (Figure 1b) was estimated (see below). Snell’s law of specular materials was used to estimate the flux loss of the reflected radiation reaching the reactor

5114

xp ) r cos θ

-bi ( √bi2 - 4aici xW ) 2ai

generating the coordinates of the points where the reflected ray could intercept the reactor wall. An imaginary solution of eqs 13-17 implied that the reflected ray intercepted the CPC surface again and did not reach the reactor wall. In such cases, the coordinates of the new point of reflection were estimated numerically by the error-function developed in eq 19

LVRPA )

For the incident flux entering the reactor through the upper hemicylindrical wall exposed to the sun (CPC and tubular reactor), simple geometry was applied to calculate the position of the ray with respect to the y-axis, xp, the total distance δ traveled by each ray, and the distance rp traveled by the ray to reach (r, θ) (Figure 1a)


The distance rp in the LVRPA (eq 21) was

]

(28)

θk,2 ) π - θ + θk,0 - θk,1

(29)

θk,1 )

r)

{

RR

[

π - (θk,F - θk,0) 2

sin θk,1 for θ * θk,0 or θ * θk,F sin θk,2 RR for θ ) θk,0 or θ ) θk,F

rp ) RR · sin |θ - θk,0 |/sin θk,2

(30)

(31)

For the reflected radiation, the radial coordinate was defined by eq 30, and rp by eq 31. The LVRPA of each point in the polar grid of the photoreactor was calculated with eq 21 summing up the contribution of the direct and reflected rays.

Materials and Methods A standard geometry of a commercial CPC reactor made with Duran borosilicate glass was considered (24). The tube radius was 1.65 cm, the CPC involute reflector acceptance angle was 90° (nonconcentrating), and the profile is shown in SI, Figure S1. The reflectivity of the collector surface was 0.85 but was varied to 0.50 and 0.15 to simulate fouling. The solar UV radiation flux I0 at the reactor location was assumed to be 30 W/m2, the average intensity achieved in a clear sunny day (27). The optical properties of a suspension of TiO2 (Degussa P25) were used. They were calculated from the optical properties of the catalyst combined with the solar radiation spectrum (22). The specific mass absorption and scattering coefficients were κ ) 174.75 m2 kg-1 and σ ) 1295.75 m2 kg-1, the scattering albedo was ω ) 0.88.

Results and Discussion Incident Radiation Flux. Figure 2 shows the results of the dimensionless intensity of the incident radiation entering the reactor wall. The values plotted between 0° and 180° correspond to the radiation that enters the reactor from

FIGURE 2. Dimensionless incident radiation flux entering the photoreactor walls. I0 is the incident solar radiation flux. Ψ is the reflectivity of the involute (CPC reactor only). The values plotted between 0° and 180° correspond to the radiation that enters the reactor from the upper hemicylindrical wall, exposed to the sun (tubular and CPC reactors). The incident radiation reaching the reactor from below by reflection only is represented by the profiles from 0° to -180° (CPC reactor only).

FIGURE 3. LVRPA profiles for 0.1 g/L of TiO2; Ψ ) 0.85. (a) Tubular and (b) CPC. Optical parameters: K ) 174.75 m2 kg-1; σ ) 1295.75 m2 kg-1, ω ) 0.88. The peculiar low values of the LVRPA near -30 and -150 in Figure 3(b) are due to the numerical discretization method and should be discarded. Smaller discretization intervals would eliminate such aberrations at the expenses of computational time. VOL. 44, NO. 13, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 4. LVRPA profiles for 0.5 g/L of TiO2; Ψ ) 0.85. (a) Tubular and (b) CPC. Optical parameters: K ) 174.75 m2 kg-1; σ ) 1295.75 m2 kg-1, ω ) 0.88. The peculiar low values of the LVRPA near -30 and -150 in Figure 4(b) are due to the numerical discretization method and should be discarded. Smaller discretization intervals would eliminate such aberrations at the expenses of computational time. the upper hemicylindrical wall, exposed to the sun (tubular and CPC reactors). The incident radiation reaching the reactor from below by reflection only is represented by the profiles from 0° to -180° (CPC reactor only). The reflected radiation (ψ ) 0.85) was up to 56% of the incident flux in the upper region of the tube (0°-180°), and it decreased dramatically with material reflectivity. It was practically negligible when ψ e 0.15 due to energy losses from ray reflection. The incident flux at the bottom half of the reactor wall also decreased (CPC reactor only); however, the radiation that reaches this region did not undergo multiple consecutive reflections before reaching the reactor wall. Therefore, the radiation loss was not as dramatic as for the top half of the tube (CPC reactor only). The integration of the incident photon flux over the surface area of the tube yields the total UV solar power received by the reactor per unit length, which is necessary for the estimation of the apparent quantum efficiency of photocatalytic reactions. 5116

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In contrast, the total radiation intensity at the reactor wall includes contributions from both the external incident radiation flux and the radiation scattered in the reactor. This can be calculated simply by I(RR,θ) ) LVRPA(RR,θ) /(κ × ccat)

(32)

where κ is the spectral averaged specific mass absorption coefficient (SI), and ccat is the photocatalyst concentration. Since the backscattered radiation increases with catalyst concentration (21), the total radiation intensity at the reactor wall can be significantly higher than the incident radiation, but the radial gradient of the LVRPA will be sharper. Radiation Absorption in Tubular and CPC Reactors. Effect of Catalyst Loading on the Spatial Distribution of the LVRPA. The angular and radial distributions of the LVRPA for a cross section of the solar reactors are shown in Figures 3 and 4 for both tubular and CPC reactor geometries, using 0.1 and 0.5 g/L of TiO2, respectively. The tubular reactor

shows higher LVRPA values in the narrow region directly irradiated by sunlight and negligible irradiation in the lower region of the reactor. Conversely, the CPC reactor allowed for a more uniform irradiation of the entire cross section of the reactor, as a result of the radiation reflected by the involutes. The LVRPA distribution in both photoreactors was more uniform at lower catalyst concentration (0.1 g/L) (Figure 3) and varied significantly at higher catalyst concentration (0.5 g/L) (Figure 4). Overall, significantly higher photon absorption was observed at 0.5 g/L than at 0.1 g/L, due to higher photon absorption and scattering, at higher catalyst loading. However, at higher catalyst concentration sharper LVRPA gradients occurred, and an overall decrease of photon absorption was anticipated since photons would not reach the most inner zones of the tube. Therefore, an optimum catalyst concentration exists for a given tube diameter which maximize radiation absorption in both tubular and CPC reactors. The literature has attributed this phenomenon to a “clouding” effect due to an increasing amount of suspended catalyst particles, which blocks photons reaching the inner zones of the reactor. In this study, we go beyond such a qualitative explanation and provide a quantitative evaluation of such effect, in terms of radiation absorption and by introducing the concept of apparent optical thickness. Optimum Catalyst Loading. The radiant energy absorbed in the reactor was determined by integrating the LVRPA across the cross section of both reactors. This was represented conveniently by the volumetric rate of photon absorption (VRPA) per unit reactor length VRPA/H )

∫ ∫ 2π

0

RR

0

(LVRPA)drdθ

(33)

The VRPA is essential for the evaluation of the quantum yield of photocatalytic reactions based on the amount of radiation absorbed by the catalyst. This is more significant information compared to the quantum efficiency which is based on the incident photon flux. It also allows for an appropriate evaluation of the activity of catalysts exhibiting different optical properties. Figure 5 (main) shows the presence of maxima for the VRPA/H in both tubular and CPC reactors, at specific catalyst concentrations. The catalyst concentration that optimizes the absorption of photons varies with optical properties of the catalyst in suspension (scattering albedo) and also with tube diameter. The decrease of VRPA/H is due to both scattering of photons (mainly backscattering) and the reduction of the actual volume of the reactor effectively irradiated by photons, at catalyst concentrations beyond the optimum. The inset in Figure 5 shows the maxima of the VRPA/H and the optimum catalyst concentrations as a function of scattering albedo, for both tubular and CPC reactors. This information is significant for photoreactor design since it eliminates the need of statistical analysis of experimental designs which implies considerable consumption of time and resources. The CPC shows about 70% higher radiation absorption compared to the tubular reactor, which agrees well with the experimental results by others (23, 28) using tubular and CPC reactors of similar diameter as in here, operating near optimal TiO2 concentration (0.2 g/L). In addition, the CPC requires 39% less catalyst than the tubular reactor to operate under optimum conditions, which implies lower operating costs in industrial applications. The inset in Figure 5 also shows that the radiation absorbed in both reactors increases at lower scattering albedos. Radiation scattering is related to the prevalent mixing conditions in the reactor, which is known to affect the size of the TiO2 particles in suspension, usually found in an agglomerated state (29). The size of these agglomerates also depends on pH (25). Malato-Rodrı´guez et al. (24)

FIGURE 5. Effect of catalyst loading and scattering albedo on the VRPA per unit of reactor length. CPC reactor (solid line) and tubular reactor (dotted lined). Inset shows the locus of maxima of the VRPA/H and the corresponding optimum catalyst loading as a function of the scattering albedo of the photocatalyst. The specific mass absorption coefficient was kept constant (K ) 174.75 m2 kg-1), and ω was varied. reported that after numerous experiments with different solar CPC reactors at Plataforma Solar of Almeria (Spain) using tube diameters similar to this work, the optimum TiO2 (P25) concentration was 0.2 g/L, which agrees with the results in Figure 5 (0.21 g/L for 0.88, the scattering albedo of TiO2 Degussa P25 in water). Other studies with solar CPC reactors of diameters as in here reported optimal TiO2 loading for pollutant removal of 0.2 g/L for 2,4dichlorophenol (30), 4-chlorophenol (31) and pentachlorophenol (32), 0.25 g/L for dichloroacetic acid (33), and 0.5 g/L for phenol (30). The experimental results reported in the above literature validated very closely the results predicted by the present model. Beside radiation transport considerations, chemical/physical aspects may affect the exact optimum catalyst concentration for a given substrate (34). Optimization of Radiation Absorption in Solar CPC and Tubular Reactors of Any Diameter. The results in Figure 5 apply to the reactor diameter selected in this study. The concepts of optical thickness and apparent optical thickness were introduced to remove this limitation and to determine universal design criteria of solar CPC and tubular reactors. Using the spectral averaged extinction coefficient β ) (σ + κ), the optical thickness τmax calculated from the tube diameter 2RR can be defined as τmax ) βccat(2RR)

(34)

This dimensionless optical parameter removes the dependence of the optimum catalyst concentration discussed above on the tube diameter. The apparent optical thickness τapp,max (see SI for definition of parameters) is a more significant design parameter, deriving from the SFM 2 τapp,max ) aτmax√1 - ωcorr

(35)

It removes the dependence of the optimum catalyst concentration on both tube diameter and scattering albedo. Figure 6 shows the dependence of radiation absorption from τmax and τapp,max as a function of the photocatalyst scattering albedo, in both tubular and CPC reactors. The optical thicknesses that maximize radiation absorption varied significantly with the scattering albedo in both reactors (Figure 6a). Conversely, the apparent optical thickness was insensitive to variations of the scattering albedo (Figure 6b). The optimum values of τapp,max were 7.78 for the CPC and VOL. 44, NO. 13, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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(in black) it shows that the optimization procedure presented here is independent of tube diameter and catalyst loading. The differences between the red and black lines (ω ) 0.85) are to be attributed to the numerical discretization method as reported in Figures 3-4. These results provide a new systematic method for optimization of solar CPC and tubular reactors, based on radiation transport considerations. They also provide a thorough method for estimating quantum yields based on the actual amount of direct solar radiation absorbed by a CPC or tubular photoreactor. In the presence of significant diffuse solar radiation the method described in our earlier work (22) should be followed to estimate the total solar radiation absorbed in the reactor. Experimental studies have shown the CPC to be a highly efficient reactor for environmental applications of solar photocatalysis. This study has quantified theoretically the advantages of using CPC photoreactors in terms of photon absorption and shown the application of the simpler SFM as a tool for the optimization of solar photocatalytic reactors. The concepts and models explained here can be easily extended to photocatalysts with different optical characteristics and in different environmental fields other than pollutant remediation such as hydrogen evolution by solar photocatalytic reforming of biomass waste (9). The accurate modeling of the radiation field in a CPC solar photoreactor by the SFM should also allows the estimation of kinetic parameters and rate equation, independent of radiation field in the reactor (22), which should be appropriate to use in photocatalytic reactors of any size and geometry operated under similar experimental conditions.

Acknowledgments Li Puma thanks NATO (Grant CPB.EAP.SFPP982835) and E.ON AG (E.ON International Research Initiative) for financial support. Responsibility for the content of this publication lies with the authors. Machuca-Martıńez and Colina-Ma´rquez thank the Vice-Head Office of Research of the Universidad del Valle (Grant 2520) and COLCIENCIAS for financial support (Grant 110647922029) and Ph.D. scholarships. FIGURE 6. Effect of the optical thicknesses τmax and the apparent optical thicknesses τapp,max on the VRPA per unit of reactor length as a function of the scattering albedo of the photocatalyst. CPC reactor (solid line) and tubular reactor (dotted lined). The specific mass absorption coefficient was kept constant (K ) 174.75 m2 kg-1), and ω was varied. The red lines refers to data from S2 (SI) obtained using different CPC tube diameters (0.4-6.0 cm) and three different catalyst loadings (0.2, 0.6, 1.0 g/L) and at constant optical parameters, ω ) 0.85 and K ) 174.75 m2 kg-1. 12.97 for the tubular reactor which are recommended for reactor design. The effect of catalyst loading and tube diameter are included in the definition of τapp,max (eqs 34-35) a key dimensionless parameter for optimal design of photocatalytic reactors and scale-up. Therefore, providing reactors have identical apparent optical thickness, either by varying the tube diameter or the catalyst concentration to maintain constant τapp,max, the reactors should perform similarly in terms of photon absorption. The red lines drawn in Figure 6a and 6b correspond to results of simulations using different CPC tube diameters (0.4-6.0 cm) and three different catalyst loadings (0.2, 0.6, 1.0 g/L) and at constant optical parameters, ω ) 0.85 and κ ) 174.75 m2 kg-1. They are drawn from the results shown in Figure S2 (SI) by estimating the corresponding τmax and τapp,max for each tube diameter and catalyst loading. When compared to the results for ω ) 0.85 5118

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Appendix A a ai, bi, ci b ccat fCPC,error H I0 L LVRPA m n pb, pf, ps r rp RR t x y VRPA z

SFM parameter, dimensionless second order equation parameters SFM parameter, dimensionless catalyst concentration, kg m-3 error function, m reactor length, m direct incident radiation flux at reactor wall, W m-2 lamp length, m local volumetric rate of photon absorption, W m-3 reflected ray slope, dimensionless number of consecutive reflections, dimensionless back, forward, and side scattering probabilities, dimensionless radial coordinate, m auxiliary coordinate in the photon flux direction, m reactor radius, m parameter angle, radians horizontal coordinate, m vertical coordinate, m volumetric rate of photon absorption, W axial coordinate, m

Greek letters R geometrical parameter, dimensionless reflection angle, radians Ri β geometrical parameter, dimensionless

βι γ δ φ λ λωcorr θ θ1 θk,0 θk,F θk,1 θk,2 κ σ τ ω ωcorr ψ

normal angle, radians SFM parameter, dimensionless distance travelled by ray within the reactor, m latitude, radians radiation wavelength, nm SFM parameter, m polar coordinate, radians incident angle, radians polar coordinate of entering photon flux, radians polar coordinate of exiting photon flux, radians auxiliary angle, radians auxiliary angle, radians specific mass scattering coefficient, m2 kg-1 specific mass scattering coefficient, m2 kg-1 optical thickness, dimensionless scattering albedo, dimensionless corrected scattering albedo, dimensionless reflectivity, dimensionless

Subscripts 0 relative to a boundary condition app apparent corr corrected CPC relative to the CPC i relative to a consecutive reflection max maximum min minimum p relative to the photon flux direction reflected reflected radiation R reactor W relative to the photoreactor wall λ radiation wavelength Vectors I j N r T

unitary vector in the x-axis direction unitary vector in the y-axis direction unitary normal vector position vector unitary tangent vector

Supporting Information Available Figure S1, reflected radiation in a CPC with a reactor radius of 1.65 cm and reactor profile; SFM model parameters; Figure S2, effect of the reactor radius on the VRPA/H in the CPC with three different catalyst loadings. This material is available free of charge via the Internet at http://pubs.acs.org.

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