Evaluation of Radiation Absorption in Slurry Photocatalytic Reactors. 1

Environ. Sci. Technol. 2000, 34, 2623-2630

Evaluation of Radiation Absorption in Slurry Photocatalytic Reactors. 1. Assessment of Methods in Use and New Proposal R . J . B R A N D I , † O . M . A L F A N O , †,‡ A N D A . E . C A S S A N O * ,†,‡ Instituto de Desarrollo Tecnolo´gico para la Industria Quı´mica (INTEC), Universidad Nacional del Litoral, and CONICET, Gu ¨ emes 3450, 3000 Santa Fe, Argentina

Photocatalytic reactions are the result of a light-activated process by which an appropriate semiconductor can generate electrons and holes that, afterward, can participate in oxidative-reductive reactions. One of the most important applications of these processes consists of the use of these catalytic systems for oxidizing pollutants contained in water and air systems. In these photocatalytic reactions, the initiation step is always a function of the local volumetric rate of photon absorption (LVRPA). Many of these reactions are carried out in water environments where the catalyst is a suspension of small size, solid particles. Then the system is heterogeneous, and evaluation of the light distribution becomes difficult due to the concomitant presence of radiation absorption and scattering. In this paper, we present a general theoretical frame for analyzing the different methods that have been proposed to evaluate the LVRPA. Using this approach, one can have a precise knowledge and evaluation of the assumptions that are used in each method and can critically discuss their validity. Special emphasis is put in the description of rigorous procedures that account for a complete solution of the radiative transfer equation. It is shown that in order to properly compute reaction quantum yields (or quantum efficiencies for polychromatic light) scattering should always be taken into account; otherwise, large errors can be introduced. The same conclusions are valid for scaling-up slurry-type photocatalytic reactors.

Introduction Photocatalyzed reactions are produced upon absorption of light of the appropriate wavelength by a semiconductor. When the energy of the incident photons is equal to or greater than the band gap energy, absorption occurs and electrons and holes are formed. They subsequently become involved in different reactions with the surrounding environment. Thus, holes can generate highly reactive OH• radicals or directly participate in oxidative reactions. In turn, electrons can combine with available oxygen (originating other reaction intermediates) or become part of a reductive reaction. A large variety of photocatalytic reactions have been studied involving different types of catalyst. The choice of * Corresponding author fax: +54-342-4559185; e-mail: acassano@ alpha.arcride.edu.ar. † CONICET. ‡ Universidad Nacional del Litoral. 10.1021/es9909428 CCC: $19.00 Published on Web 05/18/2000

 2000 American Chemical Society

catalyst has been oriented toward a family of metal oxides such as TiO2, ZnO, Fe2O3, and WO3 and other semiconductors, for example, CdS. Many different organic and inorganic compounds have been subjected to photocatalytic oxidative degradation. In the vast majority of them, complete mineralization has been reached; i.e., for many organic substrates, the final reaction products are water, carbon dioxide, and sometimes a mineral acid such as hydrochloric acid (1-4). After the different possibilities have been compared, titanium dioxide has been the preferred choice for much of the work. These reactions have been proposed as one potential advanced oxidation technology for water remediation (5, 6). The fact is that pollutant degradation in the gaseous and liquid phases is the main driving force for research in photocatalysis. Two approaches are in use: (a) the catalyst is a suspension of fine particles in the contaminated water, employing a low solid concentration, slurry reactor; and (b) the catalyst is immobilized in a support, and the reactor can take different configurations such as (i) a fixed bed (7, 8), (ii) a fluidized bed (9), (iii) an immobilized membrane fixed on the reactor walls (10-12), (iv) a reactive wall reactor (2, 13, 14), and (v) an immobilized film coating a bundle of optical fibers (15, 16). Even if not conclusively demonstrated, it has been often said that the largest catalytic efficiency is obtained with the suspended solid approach (17-20). However, it is also recognized that the catalyst separation cost in the treated water may invalidate this advantage. Despite that, much of the work in water systems, particularly for kinetic studies, is still carried out in slurry reactors, and the knowledge databasesfar away from completeshas been better developed for these systems. In this work, we have made a critical analysis of the different methods in use to evaluate the local volumetric rate of photon absorption (LVRPA) and succinctly described our own proposal for its calculation. This proposal results from a research program initiated several years ago aimed at developing theoretical, experimental, and numerical procedures to analyze and model scattering in photocatalytic reactors.

Volumetric Rate of Photon Absorption [eaλ(x,t)] The LVRPA (in units of einsteins per second and per cubic centimeter of the reacting mixture) is the local property of the radiation field that participates in a well-defined photochemical kinetic model. A photocatalytic reaction is a particular case of a photochemical reaction. As such, there will always be an initiation reaction step that leads to the activation of the involved species, in this case the solid semiconductor. Thus

TiO2 + hν w e- + h+

(1)

After activation, electrons and holes can recombine or interact with other species present in the bulk of the solid, in the semiconductor surface, or in the adjacent fluid. This initiation reaction proceeds at a given rate that is a function of the LVRPA. For monochromatic radiation, the rate of electron-hole generation can be written in terms of the LVRPA:

Rinit,λ ) Φinit,λeaλ

(2)

In eq 2, the following definition for monochromatic radiation has been used: VOL. 34, NO. 12, 2000 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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(

no. of available electrons (or holes) for recombination or chemical reaction per unit time and unit reaction volume Φinit,λ ) no. of λ photons absorbed per unit time and unit reaction volume

(

)

)

approaches in use to evaluate the VRPA in heterogeneous photoreactors.

(3)

This quantum yield does not include the electron-hole recombination reaction that is considered to be part of the general reaction scheme. Under normal conditions, one should expect this yield be close to one. Note that Φinit,λ must not be confused with the overall reaction quantum yield that is a much poorly defined property. In effect, for a reaction of the form hν

absorbing reactant (A) 98 products (B + C)

(4)

an overall reaction quantum yield is often used and defined as

Φoa,λ ) no. of reactant molecules (A) decomposed or of a given reaction product (B or C) formed, per unit time and unit reaction volume (5) no. of λ photons absorbed per unit time and unit reaction volume

(

)

(

)

Definitions given by eqs 3 and 5 are particular cases of a more general proposition in which the numerator is written in terms of “the rate of a given event” and where the word event stands for the formation or disappearance of any species. For polychromatic radiation, a quantum efficiency η has been also defined as (21)

∫

R)η

λ2 a eλ

λ1

dλ ) ηea

(6)

If the LVRPA is known, it can be used to calculate overall quantum yields (quantum efficiencies) or included in the kinetic model corresponding to an adopted reaction scheme or mechanism. Clearly, in the second case when a detailed description of the initiation reaction is explicitly included in the derived kinetic expression, the photon absorption rate must be formulated. A typical example of this application can be found in two recent papers (22, 23) where the initiation rate was included in a general reaction sequence originally proposed by Turchi and Ollis (24), which is based on the hydroxyl radical attack to the hydrogen-carbon bond of an organic molecule. Considering the different steps associated with this reaction scheme, the following form of an equation for the degradation rate of the organic compound was obtained: λ2

rs(x,t) ) u[Ci(t),Cmp,Sg,

∑Φ

a init,λeλ(x)]

(7)

λ1

In eq 7, Ci is the pollutant concentration, Cmp is the catalyst concentration, and Sg is the catalyst surface area. Either for calculating eq 7 or the overall quantum yield, the true LVRPA must be known. Several approaches have been proposed and are in use to calculate its value. This paper presents a theoretical analysis of them starting from a general and rigorous formulation of the radiation field description of the heterogeneous reaction space. The experimental values of the LVRPA in a specific photocatalytic reactor will be presented in an accompanying paper in this issue. There, a two-dimensional flat-plate reactor is used to precisely calculate the reaction volume averaged rate of photon absorption (VRPA) by measuring the radiative fluxes coming into and going out of the reactor walls. Then, these values are used to decide the validity of the different 2624

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ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 34, NO. 12, 2000

Evaluation of the LVRPA A systematic analysis of the different ways that have been used to evaluate the LVRPA can be directly obtained from the mathematical description supplied by the radiative transfer equation (RTE) written for a single and arbitrary direction of photon traveling. For photons of a given energy (wavelength, λ) and a given direction of propagation (Ω) the RTE can be written as (25)

dIλ(x,Ω,t) + κλ(x,t)Iλ(x,Ω,t) + ds absorption jλee(x,t) σλ(x,t)Iλ(x,Ω,t) ) + spontaneous out-scattering emission σλ(x,t) Ω′)4π 4π in-scattering due to multiple scattering p(Ω′ f Ω; λ′ f λ)Iλ(x,Ω′,t) dΩ′ (8)

∫

The RTE is written in terms of the spectral (monochromatic) specific intensity, a property that has units of einsteins per unit normal area, unit solid angle of propagation, unit time, and unit wavelength interval. Equation 8 says that along the distance measured by s in space, at a point x, photons may be lost by absorption and out-scattering. At the same time a gain of photons is possible. Spontaneous emission at high temperatures and in-scattering resulting from multiple scattering are sources of new photons for the wavelength and direction under consideration. In this equation, the medium is considered to be pseudo-homogeneous, and no other assumption has been made. Two parameters are required: the absorption coefficient (κλ) and the scattering coefficient (σλ). Their simple characterization asks for the assumption of independent scattering that is easily fulfilled by the catalyst concentrations normally used in photocatalytic systems. One function is needed to describe scattering: the phase function [p(Ω′ f Ω; λ′ f λ)]. Its evaluation is usually accomplished with the assumption of elastic scattering that means that after scattering no change in energy occurs; then, p ) p(Ω′ f Ω) alone, a condition normally valid in scattering by solid particles of the type used in photocatalytic reactions. This equation needs several boundary conditions depending upon the geometry and dimensions of the model adopted for the problem under consideration. In every case, the boundary condition for the incoming radiation to the reactor will be present. It is defined by the characteristics of the radiation source that is employed, for example, solar irradiation, lamp irradiation, etc. Generally, it can be represented by an equation of the form

Iλ(s ) s0,Ω ) ΩIN,t) ) I0λ(ΩIN,t) ) I0λ(θIN,φIN,t)

(9)

where θ and φ are the spherical coordinates employed to represent the solid angle Ω. In some cases, eqs 8 and 9 are sufficient to define the complete system of “integro-differential equation and boundary condition” required to evaluate the radiation specific intensity for a given wavelength at any position and direction inside the reactor. Sometimes additional boundary conditions will be required, particularly for multidimensional models. Once the specific intensity is known, integrating overall possible directions of irradiation (a solid angle for radiation transport), the monochromatic LVRPA can be readily obtained:

TABLE 1. Methods To Evaluate the LVRPA approach

description

1

radiative flux measurements at the window of radiation entrance to the reactor homogeneous actinometry inside the reactor volume homogeneous actinometry measuring also out-scattering radiative fluxes coming out from the other reactor boundaries partial application of radiative transfer equation inside the reactor volume full application of the radiative transfer equation inside the reactor volume

2 3 4

∫ I (x,Ω,t) dΩ

eaλ(x,t) ) κλ

4π λ

typical refs

(10)

With this information, we can now propose a classification of the different approaches that have been used to calculate the LVRPA (Table 1). Approach No. 1: Scattering Is Not Considered [σ ) 0 in Eq 8]. Method 1: Measuring the Inlet Boundary Condition. This method is based on the experimental measurement of the radiative flux at the reactor wall (or window) through which radiation goes in. Some type of photocell or photomultiplier with the corresponding transducer is normally used. The usual result supplied by this instrument is the radiative flux normal to the surface of radiation entrance. An average value over the whole surface area of illumination is normally obtained. Very often a similar result is obtained with an actinometric reaction under conditions of complete radiation absorption (all photons entering the system are captured by the actinometer); this method is analyzed in the next section. The meaning of this measurement using detectors can be explained according to

〈qn,λ〉0AR )

∫q

1 AR

(x AR n,λ

) x0) dA

27-29

comments normally, only a first-order approximation

30, 31 32-37, 58-60 better approximation than approach 1 38, 39, 61-64 40-52, 54

approximate; application is usually conditioned by limitations of the method accurate, but involving rather complex numerical methods of solution

measured photon absorption rate with this method includes all the relevant wavelengths of absorption by the catalyst and only those. This is a condition that cannot be always fulfilled. It is also recommended to use a high actinometer concentration such that, in practical terms, complete absorption of the incoming radiation is obtained. This permits the assumption that κλ f ∞ for all wavelengths. Nearly always, some sort of one-dimensional model is assumed, and a single value of the photon absorption rate for the whole reactor volume is also obtained. With proper experimental design, monochromatic or polychromatic measurements can be obtained. For the one-dimensional model (in theory, only rigorously valid for a collimated radiation beam made of parallel rays), the radiation intensity, the radiative flux, and the incident radiation have the same value (26):

Iλ(Ω,x,t) ) hI λ(x,t)δ(Ω - ex)

(14)

hI λ(x,t) ) qx,λ(x,t) ) Gλ(x,t)

(15)

where, in general terms, the incident radiation have been defined as (25):

(11) Gλ(x) )

∫ I (x,Ω) dΩ

(16)

Ω λ

where

qn,λ )

∫

I (x,Ω)Ω‚n ΩIN λ

dΩ

(12)

Then, the general eq 8 for a one-dimensional model, with monochromatic radiation with no emission inside the reaction space and no scattering, in terms of the incident radiation Gλ(x), takes the following form:

This normal radiative flux at the window of radiation entrance (in W cm-2 or, in photochemical units, in terms of einstein cm-2 s-1) is converted into the VRPA (a single value over the whole reactor volume) considering that after a correction for reflection the remaining photons are fully and uniformly absorbed. Then

〈eaλ〉VR

= (1 -

Γw,λ)〈qn,λ〉A0 R

AR VR

(13)

This method does not account for the optical properties of the reacting suspension inside the reactor. Obviously, scattering is not considered, and moreover, it is assumed that a single value of the ea exists over the whole reactor volume. In principle, monochromatic or polychromatic measurements can be obtained. Considering that all the radiation entering the reactor is absorbed by the photocatalyst is equivalent to assuming that the reactor operates as a blackbody; hence, the resulting value is the maximum possible photon absorption rate. Method 2: Homogeneous Actinometry. A second method has been extracted directly from homogeneous photochemistry. A chemical actinometer is used. Under the best operating conditions, the choice of the actinometer is made trying to match the wavelength absorption ranges of the photocatalyst and the actinometer. Then, the experimentally

∂Gλ(x) ) -κλGλ(x) ∂x Gλ(x ) 0) ) Gw,λ

(17)

When absorption along the reactor length becomes very high, the volume averaged LVRPA results:

〈eaλ(x)〉LR|κf∞ )

Gw,λ 1 G [1 - exp(-κλLR)]|κf∞ = (18) LR w,λ LR

For the actinometric reaction and using monochromatic radiation, a mass balance for a well-stirred, batch photoreactor gives

dCi ) m ) νiΦλ,i〈eaλ(x)〉LR dt

(19)

valid for the concentration range where the reaction rate is first-order with respect to the LVRPA. In eq 19, νi is the stoichiometric coefficient of component i; Φλ,i is the actinometer monochromatic overall quantum yield (assumed independent of the actinometer concentration in the abovementioned range) and for the one-dimensional model the volume average of the LVRPA is to be calculated over the VOL. 34, NO. 12, 2000 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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reactor length. From this value, the VRPA can be expressed by

〈eaλ(x)〉LR )

(m/νi) Φλ,i

(20)

For polychromatic radiation:

dCi dt

∑〈e (x)〉 a λ

) m ) ν i ηi (

LR)

(21)

λ

where

FIGURE 1. Variables involved in the radiation balance.

∑Φ

( ηi )

a λ,i〈eλ(x)〉LR)

λ

)

∑

(

〈eaλ(x)〉LR)

∑Φ

λ,i

λ

( ) Gw,λ Gw

(22)

λ

With this value, we can obtain

〈ea(x)〉LR ) (

∑〈e (x)〉 a λ

λ

LR)

)

(m/νi) ηi

(23)

The value provided by eq 23 is afterward used to compute the VRPA by the photocatalyst. The choice of the actinometer concentration such as to obtain complete absorption is equivalent, once more, to a reactor behaving as a blackbody. In this case, the absorption and scattering characteristics of the catalyst are also ignored. Consequently, the same drawbacks previously indicated for method 1 also apply. The computed photon absorption rate is the maximum expected value. This method has been used very often in most of the chemistry literature on photocatalytic reactions (27-31). Normally, recognizing the limitations of this approach, the result is reported as an “apparent” absorption rate. Approach No. 2: Out-Scattering Is Partially “Measured”. One interesting variation of the previously described method aimed at reducing errors in the LVRPA evaluation is to combine the previous technique with the use of photodetectors and/or actinometers placed at all the walls that define the reactor volume except for (unfortunately) that corresponding to the incoming radiation. This second set of measurements is made while using the photocatalyst inside the reactor. With this approach, these additional measurements are intended to provide the required information to compute radiation escaping from the remaining boundaries of the reactor due to scattering. Then, a radiation balance is performed subtracting from the incoming radiation the one that is escaping by scattering (Figure 1):

〈ea〉VRVR ) (qIN - qBACK)AIN - qFORAFOR - qSIDEASIDE (24) Backscattering at the wall of radiation entrance (qBACK) is difficult to account for and, unfortunately, has not been measured in these proposals. Additionally, once more, a single value for the photon absorption rate for the whole reactor volume is obtained. These experiments should be repeated every time that the catalyst type or the catalyst concentration is changed because scattering is affected by the catalyst characteristics and loading. This approach and variations derived from it have been used by refs 32-36. Sometimes the walls that define the reactor, except for that corresponding to the incoming radiation, are covered with a reflecting surface (37). In this way, some (but never all) of the out-scattered photons may be sent back into the reaction volume. This is a variation of the method described above. 2626

9


Approach No. 3: Simplified Solution of the RTE. Most of the difficulties associated with the solution of eq 8 are due to the term representing in-scattering. An approximate method for evaluating the LVRPA was proposed that considers that, under very restricted operating conditions, in-scattering could be minimized (38, 39). The approximation called “the fully irradiated photochemical reactor”sa cylinder of annular cross sectionswas achieved with an experimental device that had a very low optical thickness; this was accomplished with low catalyst loading and very short optical path. Radiation coming in and going out of the reaction space was measured with actinometers; the latter during the course of the photocatalytic reaction. Maintaining our discussion in Cartesian geometry, this type of result can be analyzed using a simple one-dimensional model. Note that out-scattering is fully incorporated. Essentially, this method reduces the RTE to

dIλ(x,Ω,t) ) -[σλ(x,t) + κλ(x,t)]Iλ(x,Ω,t) ds

(25)

Considering a one-dimensional model, multiplying eq 25 by dΩ and integrating over the solid angle, we can get the RTE in terms of the incident radiation Gλ and the corresponding boundary condition:

dGλ ) -βλGλ dx Gλ(x ) 0) ) Gw,λ ) hI λ(x ) 0)

(26)

With actinometer measurements, the boundary condition was experimentally measured. Afterward, the simplified RTE may be applied to the photocatalytic system. The LVRPA for this system results in

ea(x) )

∫κ {G λ λ

w,λ

exp[-βλ(x)]} dλ

(27)

In this case, the absorption and scattering coefficients for the titanium dioxide suspension are needed. They can be obtained according to published procedures (40). Approach No. 4: Solving the Complete RTE (Our Proposal). Two problems must be considered: (i) the knowledge of the boundary conditions and (ii) the solution of the integro-differential equation. The boundary conditions can be divided in two different types: (a) those corresponding to surfaces of radiation inlet and (b) those that apply to all other surfaces of the reactor (41, 42). For the surfaces of radiation inlet, two different approaches can be used: (i) an experimental determination by actinometry or with some radiation detecting device and (ii) a mathematical model for radiation emission. For all other surfaces, normally a mathematical model is used. Sometimes the model corresponds to reflection and transmission at the reactor wall (42), while in other cases of high optical thickness, null reflection has been assumed (41, 43-45).

intended to simulate the bench-scale prototype of a solar photocatalytic reactor. A previous model (55, 56) was adapted for this particular case in order to predict the boundary condition for radiation inlet. The whole geometry of the system was designed using an optimization program in order to have an almost uniform radiative flux (not uniform intensities) at y ) 0. The system was modeled according to

∂Iλ(x,y,Ω) ∂Iλ(x,y,Ω) +η + βλIλ(x,y,Ω) ) ∂x ∂y σλ I (x,y,Ω′)p(Ω′ f Ω) dΩ′ (28) 4π 4π λ

µ FIGURE 2. Flat-plate photoreactor. R, reactor; Rf, reflector; L, lamp. Adapted from ref 45. The complete solution of the RTE has been approached from three different viewpoints: (i) Approximate analytical solutions for very simple situations, normally assuming isotropic scattering (46-49). (ii) Monte Carlo simulation of photon transport in participating media; computed results were obtained for general operating conditions (50, 51). (iii) Solution of the RTE with the discrete ordinate method (DOM) and application to different types of reactors, in some cases with experimental validation of theoretical predictions (41, 43, 44, 52). This approach was also used to propose a method for the experimental determination of the absorption and scattering coefficients of titanium dioxide solid suspensions (40), two indispensable properties for solving the RTE. The DOM transforms the integro-differential equation of the RTE into a system of algebraic equations that are amenable to machine computation. Three different discretizations are used: (i) For polychromatic radiation using lamps with superficial emission (having a continuous emission), the wavelength interval is turned discrete dividing the whole useful wavelength interval [λ1,λn] in as many 5-nm regions as needed [(λ1,λ2), (λ2,λ3) ... (λn-1,λn)]. Sometimes discretization is directly made according to each one of the emission lines of the lamp (for lamps with voluminal emission). Then, the RTE is solved for each “monochromatic reactor”, and the results are added up to represent the polychromatic case (26). (ii) A spatial discretization that can be one-, two-, or three-dimensional according to the particular reactor under consideration. This part of the problem can be solved with central finite difference techniques. (iii) At each point in the discrete spatial mesh a directional discretization is used (considering two angles) resorting to a spherical coordinate system; this is the especial feature of the DOM that permits the accounting of the spherical nature of radiation propagation in space. A brief description of this method can be found in ref 41; however, a more detailed account is to be found in the classic book of Duderstadt and Martin (53). The method has been applied to different reacting systems: (i) a one-dimensional flat-plate reactor with an isotropic boundary condition (22, 23, 43, 44, 54), (ii) a twodimensional flat-plate reactor (42, 45) and a two-dimensional annular reactor with cylindrical geometry (41). Very briefly, the method will be illustrated with the reacting system that was later used for the experimental verification of this proposal (65). Brandi et al. (42, 45) studied the flat-plate reactor under much less restrictive conditions than previous work (22, 23, 43, 44, 52, 54). Two tubular lamps and two parabolic reflectors provided irradiation. The boundary for radiation entrance did not force the condition of isotropic intensity (as in the case of the one-dimensional reactor). Hence, the generated radiation field was not diffuse and could not be modeled with single spatial and directional variables; thus, two directional variables (θ,φ) and two spatial variables represented by the reactor depth (y) and the reactor length (x) were employed (Figure 2). This setup was originally

∫

irradiated wall Iλ(x,y ) 0,ΩIN) ) u emission by lamps and reflectors, reactor wall properties, and reflection of backscattering at y ) 0

(

)

))0 {II (x(x))L0,y,Ω ,y,Ω ) ) 0

areas with null incoming radiation

(30)

IN

λ

R

λ

(29)

(31)

IN

areas with reflection Iλ(x,y ) HR,ΩIN) ) u reflection by the reactor wall (32) at y ) HR

(

)

Boundary conditions (30, 31) indicate that there is no radiation entrance from the y-z surfaces at x ) 0 and x ) LR. Boundary condition (32) considers that the nonabsorbed and forward scattered radiation may be reflected by the reactor wall at y ) HR and that backscattering is present. Both B‚C (29) and C‚B (32) use the reflection properties of the air-glass and glass-water suspension interfaces calculated with Fresnel equations (57). Internal transmittances were also incorporated into the B‚C values. With this purpose, all corrections required by the presence of the reactor walls (that were not considered mathematical surfaces) were made using the Net Flux Method adapted from Siegel and Howell (57). Essentially, this method performs a radiation balance on each reactor wall considering reflection (and eventually refraction) at each interface (air-glass, glass-water) and the internal absorption of the glass. The details can be found in Brandi et al. (42). Equations 28-32 were solved with the DOM. The obtained intensities can be used to calculate the VRPA according to

〈ea(x,y)〉VR )

1

∫ H L R R

1

∫

H R LR

y)HR

y)0

∫

y)HR

y)0

∫

x)LR

x)0

x)LR

∑e (x,y)) dx dy ) a λ

(

x)0

λ

∑κ ∫ I (x,y,Ω) dΩ) dx dy

(

λ

λ

4π λ

(33)

Employing a very precise experimental measurement of the radiative fluxes coming out of the reactor walls (forward transmitted and backward reflected, out-scattered fluxes), it was possible to conclude that the isotropic scattering model (57) provides the best representation of radiation scattering by titanium dioxide particles. This is a very important conclusion for a precise evaluation of the photon absorption rate through the solution of the RTE. The experimental results obtained with this work confirmed the quality of the proposed model and provided the required background for the present proposal.

Assessment of the Described Methods In this section, we compare the different methods used for evaluating the volume averaged LVRPA in photocatalytic reactors, employing as a model system the two-dimensional flat-plate reactor. The results reported here correspond to VOL. 34, NO. 12, 2000 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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actinometer that produces complete absorption for all considered wavelengths) renders a result that is equivalent to that obtained with method 1; i.e., all the light arriving to the window of radiation entrance is absorbed inside the reactor.

TABLE 2 component

characteristics

reactor

made of Tempax glass length 34.0 cm width 18.0 cm thickness 1.2 cm wall thickness 0.38 cm lamps (2) Philips TLK 40/09N 310 nm e λ e 410 nm nominal input power 40.0 W total nominal output 1.93 × 10-5 einstein s-1 power diameter 3.8 cm length 56.5 cm reflectors (2) made of Alcoa Aluminum with Alzak treatment parabola constant 2.4 cm height 9.6 cm opening 19.2 cm length 54 cm

The second method measures radiation escaping from all the reactor walls, except for that corresponding to the incoming radiation. These measurements are performed with the photocatalyst suspension placed inside the reactor. It must be observed that this method also produces an important overvaluation of the VRPA; the third column shows that for 0.5 g/L the result is equivalent to those obtained with the homogeneous actinometry and, hence, about 2 times higher than those corresponding to the fourth method. Clearly, backscattering at the wall of the radiation entrance produces a very important contribution to the radiation escaping from the reactor boundaries due to scattering. Notice that errors are magnified in this reactor due to its geometric characteristics: the reactor area of irradiation is quite large as compared with the lateral surfaces limiting the reactor volume. Moreover, at the surface opposite to that of radiation entrance, the escape of photons will be always small due to the strong absorption properties of titanium dioxide.

the experimental apparatus described in details in refs 42 and 45. Its most important features are shown in Table 2. Table 3 presents a comparison of the results obtained by application of the approaches described in previous sections. The third column shows results corresponding to different operating conditions. When homogeneous actinometry is used, two different reacting systems have been analyzed: the uranyl oxalate and the potassium ferrioxalate reactions in water solution. In the case of procedures 2-4, Aldrich has been used as a typical catalyst, and three different catalytic loadings have been considered. In this work, we have used this commercial titanium dioxide because when comparing computed values of the VRPA with experimental determinations (see ref 65), it is convenient to minimize those extraneous factors that could lead to a disguised result. A typical example is fouling of the reactor window by the catalyst. It has been shown (42) that fouling by Aldrich titanium dioxide is almost negligible. On the contrary, this is not the case of Degussa P 25 to mention the most widely used titania. It can be observed that the first method leads to a significant overvaluation of the VRPA (a factor close to 2 with respect to the rigorous application of the RTE). With 0.01 M uranyl oxalate and due to a fortuitous error compensation, the difference is smaller; the reason is that the nonabsorbed photons by this actinometer (note the adopted low concentration) have been partially compensated by the nonaccounted for losses of radiation by scattering. It must be noticed that using the potassium ferrioxalate reaction (an

The method that neglects in-scattering (third method) produces results that underestimate the value of the VRPA (the computed value is approximately 7 times smaller than the result using the complete RTE). According to this method, when one photon is scattered out of the direction of flight (the direction used for the radiation balance), unless it is absorbed, it will escape from the system without any other chance of being incorporated to the absorption-scattering process that is occurring in the participating medium. Neglecting in-scattering is equivalent to assume that scattering is single (as opposite to multiple) and the integralsa source termsin the RTE (eq 8) is zero. This hypothesis leads to an underestimation of the photons that are really interacting within the reaction space. Notice that this reactor does not operate under the conditions required by the procedure described by Martıń et al. (38); i.e., the optical density of this reactor is large. The fourth method includes scattering with all its components and should produce the most reliable results. Unfortunately, solution of the RTE has some numerical complications, and application of the DOM is not a straightforward computation. Moreover, information concerning the required absorption and scattering coefficients is not readily assessable. In ref 65, the proposed method has been experimentally confirmed.

TABLE 3. Comparison of Different Methods To Evaluate the LVRPA

a

approach

operating conditionsa

(× 109) (einstein cm-3 s-1)

measuring the inlet boundary condition;

qW ) 17.1 × 10-9 einstein cm-2 s-1

14.23

homogeneous actinometry

uranyl oxalate, 0.01 M potassium ferrioxalate, 0.02 M potassium ferrioxalate, 0.15 M

9.27 14.12 14.23

homogeneous actinometry; out-scattering measurements, excluding backscattering fluxes

titanium dioxide, Aldrich, 0.10 g/L titanium dioxide, Aldrich, 0.25 g/L titanium dioxide, Aldrich, 0.50 g/L

12.66 13.71 14.02

simplified solution of the RTE; in-scattering is neglected


0.95 0.96 0.96

complete solution of the RTE


6.85 7.52 7.67

HR ) 1.2 cm.

2628

9


Conclusions

x

position vector, m

From the previous reported results, it is possible to conclude the following: (i) If accurate values of the volumetric rate of energy absorption are required (either for kinetic studies or for application to reactor design), radiation scattering cannot be neglected. (ii) Scattering effects cannot be computed without the appropriate solution of the RTE. This calculation must be made despite the intrinsic difficulties associated with the solution of an integro-differential equation. (iii) Unfortunately, information for the solution of the RTE is scarce; the existing data about scattering distribution functions and optical parameters of the suspended solid systems are still very limited. (iv) The use of approximate methods to calculate the VRPA may lead to significant errors (resulting in over- or underestimations). These errors will be translated into any reaction kinetics parameter estimation or in quantum efficiency calculations (or quantum yields for monochromatic radiation). In turn, if the kinetic results are used for scale-up purposes, they will lead to erroneous design calculations. (v) Use of the RTE is important because it provides point values of the LVRPA, a result that is very important for kinetics studies and reactor design (see ref 65).

x

Cartesian coordinate, m

y

Cartesian coordinate, m

Acknowledgments Thanks are given to Universidad Nacional del Litoral (CAI+D 017/120 and 116), CONICET (PIP 205), ANPCyT (PICT 97), and FONCYT (BID 802/OC-AR PID 22) for financial help. The cooperation of Mrs. Claudia Romani in editing this manuscript is also acknowledged. A.E.C. thanks the Humboldt Foundation and the ISFH (Hannover) for support during his stay in Germany.

Nomenclature m2

A

area,

Ci

pollutant concentration, mol cm-3

Cmp

particle mass concentration, g

cm-3

Greek Letters β

σ + κ, volumetric extinction coefficient, m-1

Γ

reflection coefficient, dimensionless

δ

delta function

η

quantum efficiency, mol einstein-1

η, µ

directional cosines, dimensionless

θ

spherical coordinate, rad

κ

volumetric absorption coefficient, m-1

λ

wavelength, m

ν

frequency, s-1

νi

stoichiometric coefficient of component i

σ

volumetric scattering coefficient, m-1

φ

spherical coordinate, rad

Φ

quantum yield, mol einstein-1

Ω

solid angle, sr

Ω

unit vector in the direction of radiation propagation, dimensionless

Subscripts AR

relative to the surface area of radiation entrance

BACK

denotes backward direction

e

relative to emission

FOR

relative to forward radiation

i

relative to species i

IN

relative to the incoming radiation

init

relative to the initiation step

LR

relative to reactor length

ea

local volumetric rate of photon absorption, einstein m-3 s-1

n

means normal to a given surface area

u

denotes a function of

oa

means overall

G

incident radiation, einstein s-1 m-2

R

relative to the reactor

HR

reactor thickness, m

SIDE

relative to lateral reactor surface

h

Planck’s constant, kg m2 s-1

VR

relative to reactor volume

I

specific intensity, einstein s-1 m-2 sr-1

w

relative to the reactor wall

hI

intensity in the unidimensional model, einstein s-1 m-2

x

relative to the x-axis.

j

rate of photon emitted per unit volume and solid angle, einstein s-1 m-3 sr-1

λ

indicates dependence on wavelength

Ω

relative to the direction of radiation propagation

L

length along the x coordinate, m

Superscripts

n

unit normal vector

0

p

phase function, dimensionless

Special Symbols

q

radiative flux, einstein m-2 s-1

rs

-1 heterogeneous reaction rate, mol m-2 cat s

R

reaction rate, mol s-1 m-3

s

linear coordinate along the Ω direction, m

t

time, s

V

volume, m3

relative to the reactor wall

〈〉

average value

‚

dot product

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Received for review August 12, 1999. Revised manuscript received March 8, 2000. Accepted March 14, 2000. ES9909428

Evaluation of Radiation Absorption in Slurry Photocatalytic Reactors. 1

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