Photoreactor Analysis and Design: Fundamentals ... - ACS Publications

Znd. Eng. Chem. Res. 1996,34, 2155-2201

2166

REVIEWS Photoreactor Analysis and Design: Fundamentals and Applications Albert0 E. Cassano,**tCarlos A. Martin: Rodolfo J. Brandi: and Orlando M. Alfanot INTEC,O U.N.L.,"and CONICET,I Giiemes 3450, (3000) Santa Fe, Argentina

A review of photoreactor engineering theory and applications, rigorously derived from chemical reaction engineering principles and radiative energy transport fundamentals, is described. The design procedure is illustrated by presenting the modeling of a continuous flow, annular reactor that is used in a predictive mode. In opder to formulate the photochemical reaction rate properly, the radiative transfer equation applied t o participating and reactive, homogeneous, and heterogeneous media is presented. The paper also describes the way by which the effects produced by reflecting surfaces may be incorporated into the rate of reaction and, consequently, into the reactor model. Likewise, a precise formulation to account for polychromatic irradiation is shown. The described methodology provides photoreactor mathematical models that permit an a priori design of a photochemical reactor. Thus, from laboratory data, the scale-up can be performed with no limitations in the size or shape of a large scale application.

Contents I. Introduction 11. Photoreactor Design 111. Reactor Modeling 1. The Reaction 2. The Reactor 3. Momentum Balance 4. Mass Balances 5. Thermal Energy Balance 6. Reaction Kinetics IV. The Rate of the Radiation-Initiated Step 1. The Quantum Yield V. Radiation Field Properties VI. The Photon Transport Equation 1. Constitutive Relationships for the Photon Transport Equation 2. Working Photon Transport Equation VII. Radiative Transfer in Participating, Reactive, and Homogeneous Media (Single, Fluid Phase Photoreactors) VIII. Radiative Transfer in Participating, Reactive, and Heterogeneous Media (Multiple Phase Photoreactors) M. Emission Models for Tubular Photochemical Lamps 1. Three-DimensionalSource with Superficial, Diffuse Emission. The E-SDE Source Model (Figure 11) 2. Three-DimensionalSource with Voluminal, Isotropic Emission. The E-VIE Source Model (Figure 12)

X Limits of Integration for the 2156 2157 2157 2158 2159 2159 2159 2160 2160 2161 2161 2162 2164 2165 2166 2166

2168

2169 2170

2170

LVR.EA When Using Tubular Lamps 1. Limits for the Independent Variable 8 (Figures 11,12, and 14) 2. Limits for the Independent Variable $ (Figure 14)

XI. Working Equations for Calculating the LVREA 1. Homogeneous Systems 1.1. Lamps with Superficial Emission 1.2. Lamps with Voluminal Emission 2. Heterogeneous Systems 2.1. Lamps with Superficial Emission 2.2. Lamps with Voluminal Emission XII. The Absorption Coefficient 1. Homogeneous Media 2. Heterogeneous Media XIII. The Coupling Radiation Attenuation-Reaction Extent XlV. Final Equations for the Example: The Annular Photoreactor XV. Solution of the Example: Predictions from the Annular Reactor Model XVI. Polychromatic Radiation Sources XVII. Modeling of Reacting Systems Employing Reflectors 1. Boundary Condition 2. Limiting Angles for the Variable 8 3. Limiting Angles for the Variable $

0888-588519512634-2155$09.00/00 1995 American Chemical Society

2172 2172 2172 2172 2172 2172 2172 2173 2173 2173 2173 2173 2174 2174 2175 2175 2179 2180 2182 2184 2184

2156 Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995

XVIII. Applications to Homogeneous Systems 1. Annular Photoreactor with Simple Kinetics 2. Cylindrical Reactor inside an Elliptical Reflector 2.1. Monochlorination of Ethane under Isothermal Conditions with Polychromatic Radiation 2.2. Non-Isothermal Polychlorination of Methane with Polychromatic Radiation 3. Multilamp -Single Tube Photoreactor with Cylindrical Reflectors. Simple Reaction 4. Multitube -Single Lamp Reactor with Circular Reflector. Simple Reaction XM. Application to a Pseudo-Homogeneous System 1. Gas -Liquid Tank Photoreactor Irradiated from the Bottom. Chlorination of Trichloroethylene and Pentachloroethane XX Applications to Heterogeneous Systems 1. Plane Slab Geometry 2. Annular Geometry 3. Additional Work XXI. Conclusions Acknowledgment Nomenclature Literature Cited

2185 2185 2186 2186

2187

2188 2190 2190 2190

2193 2194 2195 2196 2197 2198 2198 2199

I. Introduction A photochemical reaction is a process that must be preceded by absorption of radiation of the appropriate energy by one molecule. Upon radiation absorption the excited molecule can be transformed, in one or more steps, into a product or it may be converted into an intermediate species that can participate in subsequent reactions of thermal nature as happens, for example, in chain reactions. Sometimes, radiation absorption occurs in a particular molecule but definitive changes occur in others, as in the case of photosensitized and photocatalyzed reactions. Photochemical reactions are attractive because photoactivation may be very selective. This advantage is the result of a precise and controlled modification in the electronic state of a molecule by radiation absorption. This selectivity is generally combined with a typical operating condition: excitation of the reactant is performed by radiation energy of very poor heating aptitudes and, consequently, photochemical reactions do not involve high temperatures, nor are they generally required because of the involved activation mechanism.

* To whom correspondence should be addressed. +

Professor (U.N.L.) and Research Staff Member (CONICET).

* Research Assistant from CONICET.

Instituto de Desarrollo Tecnol6gico para la Industria Quimica. Universidad Nacional del Litoral. Consejo Nacional de InvestigacionesCientfficas y Tknicas.

This last condition means (i)better equilibrium conditions for exothermic reactions, (ii) more favorable conditions for operating in the liquid phase (instead of the gas phase or the use of high pressures), and (iii) minimization of undesirable byproducts. Despite these advantages, photochemical reactions are not very widely used in industrial practice. They have been adopted when (i) no alternative thermal or catalytic process is available or (ii)the manufacturing scale is small and, very often, dedicated to high added value products; then, the processing difficulties and the negative effects of the operating and equipment costs are greatly reduced. In the past, one of the reasons set forward to explain this situation was the lack of (i) suitable reactor models and design procedures and (ii) sufficient quantitative information regarding the pertinent physical and chemical parameters. According t o Cassano et al. (1967) there are other reasons for the limited use of photoprocesses in large scale operations. Among them we can quote (i) size limitations; (ii) construction difficulties, (iii)lamp operation and maintenance difficulties, and (iv) reactor wall deposits that affect radiation entrance to the reactor. Most of these difficulties are associated with the almost unavoidable use of glass or quartz parts in the required equipment. These factors have mitigated against the photo route. However, the lack of proven design methods, at least for homogeneous systems, is no longer a reason as has been indicated by Cassano and Alfano (1991) and Cassano et al. (1994a) and will be shown in this Review. In recent years, research on new methods for air and water purification has concentrated efforts in developing processes to attain the chemical destruction of contaminants. One of the concepts that has received an increasing degree of attention is the use of titanium dioxide as a catalyst for the light-induced photolysis of pollutants (Schiavello, 1985, 1988; Pelizzetti and Serpone, 1986; Serpone and Pelizzetti, 1989; Pelizzetti and Schiavello, 1991; Ollis and Al-Ekabi, 1993). Catalysis by illuminated titanium dioxide is the result of the interaction of the electrons and holes generated in the photoactivated semiconductor with the surrounding medium; thus, as a consequence of light absorption electron-hole pairs are formed in the solid particle that can recombine or participate in reduction and oxidation reactions that have as a result the decomposition of contaminants. Halogenated hydrocarbons, aromatic hydrocarbons, nitrogen-containing heterocycle compounds, hydrogen sulfide, surfactants, herbicides, metal complexes, and many other compounds have been examined, particularly in water solution. Results indicated that almost any organic and many of the inorganic pollutants produced by the electrical, electronic, agricultural, textile, petrochemical, metallurgical, and many other industries can be completely destroyed or separated (Ollis et al., 1991). Research in this area, with particular emphasis in proving the feasibility of the idea, has produced significant progress to the point that photocatalytic technologies are presently emerging in the marketplace. As has frequently occurred in other fields, technological applications have been developed notwithstanding that research has yet to succeed in developing a comprehensive and sound understanding and description of the involved phenomena. The development of a reliable knowledge base is still in its initial stages in which problems related to catalyst preparation and immobilization, catalyst chemical and mechanical stability, chemistry and kinetic networks of pollutant degradation, intrinsic reaction kinetics including effects of irradiance level and substrate competition, photocatalytic reactor design, and process integration with other water treatment tech-

Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 2157 nologies, represent some of the main weaknesses (National Research Council, 1991). Neverthelessfthis novel field seems to be one of the most promising areas of photochemical applications. Reaction engineering contributions are still very scarce in spite of the existence of sound theoretical proposals published by Stramigioli et al. and Spadoni et al. in 1978. A brief report about the state of the art concerning heterogeneous reactor studies will be presented in this paper, accounting for the progress made in the past few years. A critical and comprehensive review on the different models proposed for describing radiation transport in homogeneous and heterogeneous reacting systems was published by Alfano et al. (1986a,b). These papers provide most of the existing historical background, and this information will not be repeated here. The problem of radiative transfer in photoreactors was presented in a systematic way, for the first time, by Irazoqui et al. (1976). Since then, the increasing significance of photocatalyzed reactions by solids has added important phenomena associated with system heterogeneities in the radiation transport process that were not analyzed in that paper. At the same time, successful photoreactor modelings employing rigorous radiation theory developed in the 1970s have been reported in the past 10 years supporting the proposed theory. A general and more complete formulation of the radiation field modeling in photoreactors has been recently written by Cassano et al. (1994b) and will be described in this Review. Our main objective is the presentation of a rigorous and consistent methodology for photoreactor analysis and design based on chemical reaction engineering and radiation transport fundamentals. Consequently, other approaches to the problem based on empirical approximations, even if useful for some particular application, have not been included within the scope of this Review.

11. Photoreactor Design Research in photochemistry has been, and still is, very important, and consequently, studies of photochemical reactions have continued a t a sustained rate over many decades. Product formation, reaction kinetics and reaction mechanisms have been the main objectives. Almost 40 years ago the pioneer paper written by Doede and Walker (1955) opened the route for a complementary activity centered in photoreactor analysis and design. The main ideas behind this different approach t o the problem are based on the recognition that more likely commercial, plant size photoreactors operate under nonuniform concentrations, temperatures, and-particularly important-radiation (light) distributions. In other words, point values of the reaction rates are intrinsically not uniform and in many cases are utterly different from the global ones. In some cases good mixing can be achieved and nonuniformities in concentrations and temperature may be less severe. However, under no circumstances can stirring render the photon distribution uniform if geometric effects (distance from the radiation source to each different point inside the reactor, particularly for noncollimated beams of radiation) or attenuation effects (due to radiation absorption by the intervening species) have produced a spatial distribution of the absorbed photons. Hence, light nonuniformities are irreducible. Obviously, additional complexities are encountered when the radiation-absorbing species have also spatial nonuniformities due to concentration gradients.

It may be convenient to revise what is meant by photoreactor design. In a recent publication, Cassano et al. (1994a) have pointed out that reactor design is a complete procedure defined in the following manner: Given some production specifications such as daily throughput, inlet and outlet (or initial and final) concentrations, and specifications about selectivity (if applicable), the reactor design output must provide (i) requirements for raw materials, (ii) operating conditions (single or multiphase system), type of hydrodynamic operation (continuous flow, batch, semibatch, etc.) and recycling (if applicable), (iii) number of reactors and reactor geometry (shape and dimensions), (iv) reactor flow rate or reaction time, (v) reactor temperature(4, pressure(s), and catalyst load (if applicable), (vi) requirements to avoid inhibitions and explosions, (vii) mixing requirements (if applicable), (viii) heating and or cooling requirements, (ix)reactor construction materials, (x) reactor control, safety requirements, and maintenance routines, and (xi) waste disposal specifications and other environmental controls. When dealing with photochemical reactions, the design must include the following: (1)the radiation source (lamp) specifications (output power and its spectral distribution, shape and dimensions, operating and maintenance requirements); (2) the geometrical arrangement of the radiation source relative to the reaction space; (3) the radiation entrance system into the reactor (mode, construction materials, and cleaning procedures); (4) the reflector characteristics if applicable (number, arrangement, shape, and dimensions). Since lamp sizes and some of the associated reactor specifications cannot be changed in a continuous fashion, the design always implies an iterative procedure. 111. Reactor Modeling The modeling of a reactor based on first principles requires the solution of the momentum, thermal energy, and multicomponent mass conservation equations. In the case of a photoreactor, the photon balance (radiation energy) must be added. The radiation balance can be treated separately from the thermal energy balance because thermal effects of the photochemically useful energy are generally negligible and, consequently, one is mainly concerned with the kinetics effects of the employed radiation. The used energy for the majority of the photochemical reactions corresponds to a wavelength that falls in the range 200 nm I A I 600 nm; this part of the electromagnetic spectrum is very effective for producing reactions and very ineffective for heating. Infrared radiation, if eventually produced by the lamp, is usually eliminated by the lamp cooling devices. Also, since most photochemical reactors never reach very high temperatures, generally, radiation emission inside the reactor can be neglected. These two considerations together permit the uncoupling of both balances. This asseveration does not mean that thermal effects in photochemical reactions do not need t o be taken into account, and special attention must be exercised in highly exo- or endothermic reactions. In these cases the need for maintaining the temperature within prescribed bounds turns the analysis of thermal effects into a very important part of the design. It is appropriate to indicate, however, that temperature effects on the kinetics of some photochemical reactions may not be so important as a consequence of the involved mechanism in the reaction initiation, but in many other cases the subsequent dark


(thermal) reactions may be strongly influenced by temperature. Strictly speaking, the mass (one for each reacting species) and energy balances must be written for monochromatic radiation. One would have a set of different mass and energy balances, one for each wavelength within the reactant radiation absorption wavelength interval. In practice, one assumes that the kinetics effects of the absorbed photons at different wavelengths are additive and writes only one mass balance for each species and one energy balance, adding up each of the monochromatic contributions (see section XVI: Polychromatic Radiation Sources). As usual, the mass and energy balances are coupled through the reaction rate terms, and in a photochemical reactor this coupling extends to the radiation balance; i.e., also the mass and the photon balances are coupled. This last coupling is a consequence of the effect of the radiation-absorbing-reactantconcentration on the radiation absorption by the reacting medium. Since in general this concentration changes with the reaction extent, the spatial (andor temporal) absorption of radiation changes in a similar manner. There is only one practical exception to the mass-radiation coupling and hence to the simultaneous solution requirement: a system where the chemical species that absorbs radiation and participates in the reaction keeps its concentration constant, as in the case of a photosensitized or a photocatalyzed reaction. When this is not the case, and particularly if the reactor is not perfectly mixed (for example in a continuous flow, tubular reactor), the link between the attenuation of radiation and the extent of the reaction greatly increases the complexity of the design as will be shown later (section XIII). In all but a perfectly mixed reactor the solution of the momentum balance, generally uncoupled from the mass and energy equations, will be required. This situation imposes on photochemical reactor design the same restrictions with regard to an a priori design as in any other type of reacting system. Turbulence and multiphase flows are still areas where the lack of fundamental equations of general and proven validity for predicting the flow behavior in practical problems is most critical. As a direct consequence, the modeling of mass and energy fluxes under these conditions has the same flaws in photochemical systems as in the whole area of chemical reaction engineering. In order t o keep the extension of this work within the required limitations, we will illustrate the conservation equations for the case of a steady state, continuous flow, annular photoreactor where the photochlorination of methane in the gas phase is conducted (Figure 1). Leaving aside the flat plate reactor, annular reactors constitute one of the simplest cases for evaluating the radiation field; at the same time they represent a configuration very often encountered in industrial practice. The homogeneous system is irradiated only with direct radiation (i.e., there are no reflectors) by means of a tubular lamp of the arc type, with superficial emission and monochromatic light. The extension to cases of polychromatic radiation and reactors using reflectors (with direct and indirect irradiation) will be treated in separate sections below (sections XVI and XVII). A complete and precise derivation of the conservation equations for multicomponent systems can be found in the book written by Slattery (1978). To avoid the inclusion of uncertainties in the expressions for the mass and energy fluxes, the flow will be assumed laminar and completely developed. Similar consider-

‘

a

Figure 1. Geometry of the continuous flow, annular photoreactor. [Reproduced from Cassano et al. (1994a) with permission. Copyright 1994 Research Trends.]

ations will be made with regard to simplificationsin the modeling of the reactor cooling; one could have added to the model equations a complete heat transfer problem comprising the coupling of the cooling fluid temperature changes with the reactor temperature evolution or assume the simplifying condition of constant heat transfer fluid temperature. Since the key features of the photoreactor design are not affected, we have opted for the latter. 1. The Reaction. The chlorination of methane has been chosen as the reaction to illustrate the method because it is a well-known system and it provides almost all the important features that one could encounter in a homogeneous photochemical process. The mechanism of the reaction is well established, and it involves both chlorine atoms and organic free radicals as alternate chain carriers. The mechanism of the photochemical chlorination of methane can be written as indicated in Table 1. In reaction R-1 activation (the initiation step) is produced by radiation absorption. The way to formulate the rate of initiation will be described below (section IV). Principal products (chloromethanes) result mainly from chain propagation reactions while secondary products (chloroethanes) result from the free radical-free radical termination reactions. In photochemical chlorinations the formation of byproducts such as unsaturated compounds is not important. Besides, the reaction products are free from carbon and tar deposition. In the homogeneous deactivation of chlorine (reaction R-14)a third-body collision is necessary from energetic considerations. The third body may be an inert or some of the reactants in the mixture. The substitution reactions are highly exothermic [from -9.1 x lo4 to -10.4 x lo4 J mol-l at 298 K and from -13.4 x lo4 to -14.9 x lo4 J mol-l at 623 K], and consequently, reaction heat effects may be important. De Bernardez and Cassano (1986) studied this reaction mechanism using the detailed kinetic information provided by the classical work of Noyes (1951), Chiltz et al. (19631, Clyne and Stedman (19681, and Kurtz (1972). They arrived at the conclusion that (i)neglect-

Ind. Eng. Chem. Res., Vol. 34,No. 7,1995 2169 Table 1. Reaction Mechanism

Table 2. Reaction Kinetics Constants

Initiation Step Cl2 5 2cl' Propagation Steps CH4 Cl' CH3' HCl CH3' Cl2 CH3C1+ Cl' CH3Clf Cl' CH2C1' HC1 CH2Cl' Clz CHzClz Cl' CHzClz Cl' CHC12' HC1 CHC12' Cl2 CHC1.q Cl' CHC13 Cl' * CCl$ HC1 Cc13' f c12 cc4 cl' Homogeneous Termination Steps C1' CHs' CHsCl C1' + CHiCP CH2Cl2 Cl' CHC12'- CHCl3 c1' f cc13' cc14 Cl' Cl' M Cl2 M CH3' CH3' CH3-CH3 CH3' CH2C1' CH3-CH2Cl CH3' CHC12' CHs-CHClz CH; cc13'i CH3-CC13 CH2Cl' CH2Cl' CH2Cl-CH2Cl CHzCl' CHC12' e CH2Cl-CHC12 CH2Cl' CCl3' CH2Cl-CC13 CHC12' CHC12' CHC12-CHC12 CHC12' cc13' CHC12-CC13 CCl$ CCl3' CCl3-CC13 Heterogeneous Termination Steps CH3' wall products CH2Cl' wall products CHC12' wall 4 products cc13' wall products Cl' wall products

-- + - ++ + + + + + + +

+ +

+ + + + + + + + + + + + + + + + + + +

5

6 7 8 9 10 11 12 13 14

+

---- + -3 ,

r3

0

---

(R-10) iR-llj (R-12) (R-13) (R-14) (R-15) (R-16) (R-17) (R-18) (R-19) (R-20) (R-21) (R-22) (R-23) (R-24) (R-25) (R-26) (R-27) (R-28) (R-29)

ing heterogeneous-at the wall-termination steps (reactions R-25 to R-291, the concentration profiles for the stable species are not affected, not even a t points close to the reactor wall; the differences in radial and axial concentration profiles computed with respect to the case in which reactions R-25 to R-29 are included being less than 0.1%; (ii) the incidence of homogeneous termination reactions that lead to the formation of chloroethanes (reactions R-15 to R-24) is negligible when their concentrations are compared with those corresponding to the main products, the differences being 3 or more orders of magnitude; also, they have shown that, neglecting these reactions, the output concentration of the stable, principal products remains unchanged; and (iii)the local or microscopic steady state approximation (mssa) can be safely applied to the highly reactive intermediates that are the organic free radicals and atomic chlorine. Whenever the average residence time is greater than 5 s, the molar distribution of products obtained &r application of the mssa do not differ from the case in which the full set of differential equations is solved; this time may be even smaller (tenths of a second) if the reactor is irradiated with a lamp of high output energy (Cabrera et al., 1991a,b). These simplifications permit reduction of the mechanistic sequence from 29 to 14 steps (reactions R-1 to R-14). The full set of 14 kinetic constants (preexponential factors and activation energies) were corroborated in a continuous flow reactor by Cabrera et al. (1991a), and the final results are illustrated in Table 2. 2. The Reactor. As depicted in Figure 1, the simplest practical photochemical reactor will be modeled. This methodology has been precisely verified in an annular reactor by De Bernardez and Cassano (19851,but many other reactor configurations of greater complexity have been modeled and experimentally

a

E (J mol-l)

log Aa 14.095 13.545 14.175 11.971 13.996 12.291 13.739 12.285 13.87 13.68 13.68 13.68 14.345b

step 2 3 4

Units for A: cm3 mol-'

5-l.

15 502 10 104 12 322 12 242 12 475 17 342 13 411 25 089

-7 029

Units for A: cm6 mol-2 s-l.

validated as will be briefly described below (section

XVIII). The basic ideas were presented for the first time in a research paper published by Irazoqui et al. (1973). The reactor consists of two concentric cylindrical tubes, the interior one being as transparent as possible to the employed radiation. A tubular lamp is placed at the central axis of the system, and it is refrigerated by means of a coolingjacket. Reactants and products flow through the annular space. The full reaction domain is irradiated by the lamp, but it should be expected that this irradiation will not be uniform in space. Besides, notice that the reactor length is usually longer than the lamp length. 3. Momentum Balance. The system will be studied under the following assumptions: (i) steady state, (ii) unidirectional, incompressible, continuous flow under fully developed, laminar regime, (iii) Newtonian fluid, (iv) azimuthal symmetry, and (v) constant physical properties. Perhaps the fully developed flow and the constancy of the physical properties are the two more questionable hypotheses. They may be relaxed at the expense of additional computer processing time. Then, the solution of the Cauchy's first equation of motion produces the velocity profile. In dimensionless form

where

0 ' (

(u,) =

- 'L, + em&LR)rRo 8&R

2

1 - x4 (1 - X 2

1- x2

)

ln(l/X>

(4)

and the other variables are defined in the Nomenclature. Under normal conditions (u,) is directly obtained from the volumetric flow rate and the reactor cross sectional area. 4. Mass Balances. The mass balances may be written under four more assumptions: (i) negligible axial molecular diffusional fluxes when compared with the convective flow, (ii) constant diffusivities, (iii)nonpermeable reactor walls since wall reactions were neglected, and (iv) monochromatic operation. The constancy of the diffusion coefficients only simplifies the computational work, but there are no difficulties in using the Stefan-Maxwell relationships (at any point in the reactor) as was done by De Bernardez and

2160 Ind. Eng. Chem. Res., Vol. 34, No. 7,1995

Cassano (1986). Temperature and concentration effects on the different physicochemical properties can be incorporated in a similar manner. Extension to polychromatic radiation can be done as will be explained along this paper. To simplify notation, let us identify the chemical species using the following subindexes: 1, ClCH3; 2, c12CH2; 3, c13CH; 4, q14C; 5 , Cl,; 6, CH4; 7, C1H; 8, N2; 1, ClCH2.; 2, C12CH; 3, C13C'; 5, CP; 6, CH3*;nitrogen is an inert. Then, for each stable species, the dimensionless material balance expression is U(y) av, - -- a ( y T ) = Q i , i=1-7 ag (Ge Pei) Y ay

i = 1-7

--(C,l) =0 aY

(7)

t=-

(9) In eq 9 the subindex r represents any of the intermediate, unstable species and Qw,r are the heterogeneous, a t the wall, reaction rates. 6. Thermal Energy Balance. The following additional assumptions are necessary: (i) negligible viscous dissipation, (ii) negligible axial conduction as compared with the axial convective flow, (iii) constant thermal conductivities (can be relaxed), and (iv) constant temperature for the reactor refrigerating fluid (just for simplicity). We must recall that it has been said that, additionally, thermal effects of the absorbed radiation will not be considered. Then, in dimensionless form

T - To

To - Tc

(15)

In eq 14, the values of (A.l$.R'$ are computed for the following stoichiometric equations:

+ C1, ClCH, + C1, CH,

It can be remarked that if the heterogeneous (at the wall) termination reactions of the unstable, intermediate species would have not been neglected, the mssa cannot be applied. Then, mass balances equations as the one represented by eq 5 m y t be written for every chemical species (i = 1-7 and 1-6) and the boundary conditions for unstable species should recognize that the walls act as a sink for them according t o a prescribed kinetics. The mass fluxes in the radial direction must be made equal to the reaction rates at the wall. The problem was treated by De Bernardez and Cassano (1986)and ClariP et al. (1988). For instance, one of such boundary conditions should have been written as follows:

with the following initial condition:

and

(6)

For the stable species, the boundary conditions are

av,

In eq 10 Q is given by

(5)

Reaction rates (Q) for eq 5 will be given after treating the thermal energy balance. Equation 5 has the following initial condition:

Y,(O,y)=

The boundary conditions are

-

+ C12 C1,CH + C1,

Cl,CH,

+ HCl C12CH2+ HC1

ClCH,

+ HC1 C1,C + HC1

C1,CH

-

(K-1) (K-2) (K-3) (K-4)

6. Reaction Kinetics. Equation 5 requires information about S2i for the stable species (i = 1-7). It can be obtained from the mechanistic kinetics in terms of the known kinetic constants (Table 2), the concentration of the stable species, and the concentration of the unstable species according to the mass action law. From the definitions given by Table 3 one can write the reaction kinetics as follows:

n = A-V

(16)

From Table 3 it can be seen that the solution of eq 16 requires the knowledge of 12 concentrations. However, we have only seven differential equations; the remaining five are provided by the microscopic steady state approximation (mssa) for intermediate species (atomic and free radical species). Application of the mssa renders the results provided by eqs 17-21 indicated in Table 4. At each point inside the reactor we must solve the set of equations given in Table 4. With the obtained values of the intermediate species local concentrations (five values), we have reduced our unknowns to seven concentrations for which we have seven reaction rates (Table 3) to be substituted into the seven partial differential equations of the mass balances (eq 5). Clearly, wall reactions could have never been incorporated into the point equations derived from the mssa for the homogeneous phase; they can be treated only as boundary conditions; i.e., heterogeneous reactions (at the wall) are not compatible with the mssa in the bulk of the reactor.

Ind. Eng. Chem. Res., Vol. 34, No. 7,1995 2161 Table 3. Arrays for the Reaction Rate Expressions name symbol dimensionless reaction rates dimensionless concentration array

P

dimensionless propagating reaction rate array

A

definition

[Pl a2 % n4 n5 [Y$ Y71T

%IT

Table 4. Results of the mssa

Y- -

K,Y2 + K7’Y3 + K7Y5+ Kl2YgY6

- K,‘Y7

- Emission

(19) (20)

K2Y6+ KiYl Ye - KiY7+ K3Y5+ KloYg,y5

(21)

In eq 17 Qinit is the dimensionless rate of initiation which is given by &. . =-Rini&’R lntt

REACTION RATES

- Absorption

K4Y, + KiY, Y- - K,‘Y7 + K5Y6+ Kl,Ybyg,

(22)

(V,)C,,O

is the distinct characteristic of a photochemical reaction. It will be treated in what follows.

Rinit

IV. The Rate of the Radiation-Initiated Step When expressing the rate of a photochemical reaction, it is necessary to make the distinction between dark and radiation-activated (lighted) steps. To treat the dark reactions one uses the same methodology as for conventional reactors; the main difference appears when evaluating the rate of the radiation-activated step. The existence of this very particular step constitutes the main distinctive aspect (and the most important one) between thermal (or thermal catalytic) and radiationactivated reactions. In a chain reaction it is represented by the initiation step. When presenting the quantum yield concept, it will be seen that the rate of the radiation-activated step (rras) is proportional to the absorbed, useful energy through a property that has been defined as the local volumetric rate of energy absorption (Irazoqui et al., 1976). The LVREA ( e 3 represents the amount of photons (in units of energy for a given frequency interval) that are absorbed per unit time and unit reaction volume. The LVREA depends on the radiation field (photon distribution) existing in the reaction space; hence, we must know the radiation field within the photoreactor. This radiant energy distribution is not uniform in space due t o several causes; among them, the attenuation produced by the species absorption is always present. Additional phenomena, usually also

I REACTOR AND LAMP

-4

INITIATION RATE

I

L.V. R. €.A.

I

Figure 2. Evaluation of the local volumetric rate of energy absorption. [Reproduced from Cassano et al. (1994a) with permission. Copyright 1994 Research Trends.]

important, are the physical properties and geometrical characteristics of the lamp-reactor system. The value of the LVREA is defined for monochromatic radiation, but it can be extended to polychromatic fields by performing an integration over all useful wavelengths. The “useful wavelength” concept will be treated in detail in a specific section of this paper (PolychromaticRadiation Sources). Returning to the original problem (the rras), its general structure may be described as schematically indicated in Figure 2. As was shown before, the mass balances ask for expressions formulating the reaction rates; be it a molecular or a free radical reaction mechanism, always some of the steps (generally only one) are initiated by radiation absorption. The radiation-activated-step kinetics is always written in terms of et. The evaluation of the LVREA is performed stating first the general radiation transport equation that requires the appropriate constitutive equations for absorption, emission, and scattering. The resulting radiative transfer equation is then successively applied to the reaction space where there is only absorption (in homogeneous media) or absorption and scattering (in heterogeneous media), and t o the lamp where emission is the prevailing phenomenon. Combining both results one can obtain, in a straightforward manner, the point (local) value of the rate of radiation absorption. A typical parameter in writing the rras (initiation rate for the chain reaction) is the quantum yield. It will be discussed below. 1. The Quantum Yield. Any photochemical reaction begins with the absorption of a photon by a molecule that gives rise to some form of an excited state. This initial process of activation is k n o w n as the primary event. However, after absorption of radiation, other processes different from the desired reaction can

2162 Ind. Eng. Chem. Res., Vol. 34,No. 7, 1995 Table 5. Definitions of Radiation Properties property name energy radiant energy energy per unit time radiant power energy per unit time, unit area specific intensity and unit solid angle about the direction of propagation energy per unit volume radiant energy density energy per unit time and unit area energy per unit time and unit area in a given direction energy per unit time and unit area from all directions

radiative flux vector

definition

symbol

units

J W W/(m2sr)

dEJdt dEd(dA cos e) dt dQ1

J/m3

s:

W/m2

net radiative flux

R

4”

W/m2

incident radiation

G”

W/m2

occur; for example: a different, parallel reaction, fluorescence, phosphorescence, deactivation by physical quenching, etc. Each one of them defines different paths that may occur with a given efficiency (yield)with respect to the absorbed energy. The definition of this yield depends on the nature of the process concerned. The primary process is usually defined including both the initial act of absorption and those processes following immediately after which are determined by the properties of the excited electronic state, but it excludes any form of secondary reactions or processes occurring after the first one. In many cases the primary process leads, among other primary processes, to a dissociation of the absorbing molecule; in others, such as in a photocatalytic reaction, it leads to the formation of an electron-hole pair in the semiconductor particle. Recombination of the dissociated molecules or of the electron-hole pair are not included as part of the primary event. It has been well established that, in a single photon absorption process, the rate of the radiation-activated step (primary event) is proportional to the rate of energy absorbed (the local volumetric rate of energy absorption). The proportionality constant is the primary reaction quantum yield (Noyes and Leighton, 1941; Calvert and Pitts, 1966). For the case of the initiation reaction

(23) In general, the primary quantum yield may be defined as - number molecprim (24) @prim,v - number phot, absorb. In eq 24 number molecprimis the number of molecules following the expected path in the primary process (in our case the activation that leads exclusively to chemical reaction) and number phot, absorb. is the number of absorbed quanta of radiation. Clearly, in the general case, ”prim,v is a function of frequency and its units are molecules per quanta or moles per einstein. According to the second law of photochemistry the absorption of light by a molecule is a one-quantum process, so that the sum of all primary-process quantum yields must be less than or equal to one (Calvert and Pitts, 1966). To illustrate this situation, we can consider the photochemical dissociation of chlorine molecules in the gas phase: referring to the potential curves for chlorine they unequivocally show that the upper states are unstable and the molecule dissociates very rapidly. Then this process will predominate over all others and we can say that, almost independently of the frequency considered, for chlorine dissociation, @prim,v 1. A more detailed analysis has been presented by Clariti et al. (1988).

A some practica, app,cations a I fferent definition of the yield is used. It is called the overall quantum yield. This is a highly “process-dependent” value and does not have the quality of an intrinsic kinetic property. Its definition is - number molec,,, (25) @overall~Y- number phot, absorb. In eq 25 number molecfin is the number of molecules of a given reactant or product finally disappeared or formed, respectively. Its evaluation comprises the primary event and all the other secondary reactions that follow thereafter (Noyes and Leighton, 1941;Calvert and Pitts, 1966). Generally, this property will give an indication of the nature of the reaction mechanism. If @overall,” is large, a chain mechanism is indicated; if it is small, either deactivation or important recombination is suggested, although other possibilities must not be overlooked. It can be clearly seen that while @prim,, can never be larger than unity, @overall,” can attain very large values. (Depending on the chain length it can be as large as lo6 or even more.) As indicated above, the key variable is the LVREA. From Figure 2, to evaluate e: we must know the radiation field inside the reactor. Following Irazoqui et al. (1976)and Cassano et al. (1994b),the background theory will be presented in the next sections.

V. Radiation Field Properties Radiative transfer in participating media is a wellestablished subject in the fields of physics and engineering science (Vincenti and Kruger, 1965; Pai, 1966; Ozisik, 1973;Siege1 and Howell, 1992;etc.). Concepts are well understood, properties are clearly defined, and units stick to the SI system. Nevertheless, the subject of radiative transfer in participating and reactive media has been treated with different approaches in the classical literature of chemistry and chemical physics. Most of these approaches use the name and symbol of “light intensity” (but not the units) for a property that is not a radiation intensity; others use all the concepts and definitions taken from the field of photometry but, unfortunately, the units (lumen, candle, etc.) are related to the SI system in a very odd manner; finally other groups have adopted a terminology taken from the field of radiometry that is very close to the one used in radiative transport [see for example Braun et al. (198611. In the area of photoreador engineering we have adhered to the existing ideas in engineering science. It seems appropriate to revise the most important definitions together with their names, notations, and units. They are listed in Table 5. Rigorously speaking, all properties

Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 2163

na,” - photons

Figure 3. Characterization of the radiation field. [Adapted from Pai (19661.1

are defined for monochromatic radiation. Properties for monochromatic radiation (radiation in the frequency interval between v and v dv) are expressed per unit frequency interval and are called spectral properties (for example, spectral specific intensity). All properties in Table 5 are defined per unit frequency interval. All units in joules or watts can be converted into einstein or einstein per second with the proper transformation: L1.1960 x lO-l/A (m)lW s = 1 einstein. Some of these properties deserve a short explanation. When the shortest characteristic length of the system (for instance, the diameter of a photocatalytic particle) is large compared with the wavelengths of the used radiation and the characteristic time (for example, some characteristic reaction time) is also large compared with the range of times associated with the frequencies of the radiation waves, radiation phenomena may be analyzed from a macroscopic point of view. Then, propagation of photons may be represented by bundles of rays with a given energy. These rays may be specified by the spectral specific intensity that is the fundamental property for characterizing radiation fields. In Figure 3, let dA be an arbitrarily oriented small area about the space coordinate x, P a point in this area, and n the normal to the area at point P. At a given time there will be radiation rays traveling this area element in all directions. Energy may be transmitted through, emitted by, or reflected on this elementary surface. Let us consider a specific direction along which we draw a line that is characterized by the unit direction vector 8 that makes an angle 8 to the normal n. The vector S2 coincides with the axis of an elementary cone of solid angle dS2. All elementary solid angles corresponding to rays parallel t o the direction S2 passing through dA define a truncated semi-infinite cone do, whose crosssectional area perpendicular to S2 at the point P will be dA cos 8. Let dE, be the total amount of radiative energy passing through the area dA inside the cone d o in the time dt and with an energy in the frequency range betweeri v and v + dv. The spectral specific intensity is defined as

+

(26)

According to eq 26 the spectral (monochromatic) specific intensity is the amount of radiative energy (in energy units) streaming through a unit area perpendicular to the direction of propagation Q, per unit solid

Figure 4. Characterization of the photon distribution in directions and frequencies. 7’is a volume in space, bounded by a surface A having an outwardly directed normal n. [Adapted from Whitaker (19771.1

angle about the direction 8, per unit frequency about the frequency v, and per unit time about the time t. In photoreactor engineering, the usual units for I, are einstein per square meter, per steradian, per unit frequency (or unit wavelength), and per second. The spectral specific intensity is related t o the photon density number as follows. Let us fur our attention on those photons having a direction of propagation within the differential solid angle dS2 centered about the direction of propagation $2 and a frequency range between v and v dv (Figures 3 and 4). A photon distribution function f~ can be defined in terms of the number of such photons per unit volume, so that this number is given by

+

dnR,, = nR(x,t)fR(x,t,Q,v)dQ dv (27) In eq 27 nRis the photon density number. In Figure 4 the streams of 8 , v photons are represented by thick arrows with black dots; they have (Santarelli, 1983) (i) zero mass and charge, (ii)a frequency v or a wavelength 1 = c/v, (iii) an energy hv, and (iv) a direction of propagation 8. Clearly, the spectral specific intensity is related to the photon density number by

Iv= c(hv)nRfR= c(hv)nQ,,

(28) As is known, the energy of one photon of frequency v is given by hv and c is the speed of such a photon (speed of the light). There is no fundamental difference between I, and (cnRfR),(hv) being a unit conversion factor. In the most general case, radiation may be arriving a t one point inside a photochemical reactor from all directions in space. For a photochemical reaction to occur this radiation must be absorbed by an elementary reacting volume (a material point in space);thus, pencils of radiation coming from all directions must cross the whole elementary surface that bounds such an element of volume. Consequently, the important photochemical property is the spectral incident radiation given by

G, =

sQIv dQ

(29)

In eq 29 an integration for all possible directions 8 over the entire spherical space has been performed. In a spherical coordinate system located a t the point of incidence (the reacting elementary volume) the arriving incident radiation is

G, = ~

~ sin 8 d@ ~de

~

I (30)

,


[el, 021 and [#I,421 are the integration limits that define the space from which radiation arrives at the point of incidence. If radiation energy arrives from the whole 4n space, then the limits for 8 extend from 0 to n and those for 4 extend from 0 to 2n. For each point of incidence, in practice, these limits are defined by the extension of the lamp (section X). For polychromatic radiation, an integration over the frequency range of interest must be performed (accounting for the overlapping frequency regions of lamp emission, reactor wall transmission, and radiationabsorbing species absorption coefficient):

~ ” “ h ~sin&8 dq5~d8I ,dv

G=

v1

(31)

In the elementary volume of radiation absorption, for a single photon absorption, energy is absorbed according to e;(x,t) = K , ( x , ~G,(x,t) )

(32)

e: is the spectral (monochromatic)local volumetric rate of energy absorption (LVREA) or the spectral absorbed incident radiation, very often improperly called “absorbed intensity”. Its units are einstein per cubic meter and per second, very different from those corresponding to an intensity. K is the volumetric absorption coefficient that is always some function of the concentration of the absorbing species. The well-known Beer’s approximation can be used for homogeneous, dilute systems. For polychromatic radiation

ea = A ~ K , Gdv , ea = ~

~

~

&

~

Ksin,

8Jdq5, d8 dv

(34)

M. The Photon Transport Equation According to Santarelli (1983)we can think of radiative transfer in participating reacting media as the result of the interaction of a material multicomponent continuum with an immaterial photon phase. Both phases coexist in a given region in space and interact among them according to modes that are defined through the constitutive assumptions done for each medium. We wish to write a photon balance. Following Whitaker (1977),let us consider a fured volume in space V such as that represented in Figure 4. We are interested in those photons having a flight path lying within the solid angle of propagation dSZ about the direction Q and a t the same time transported by an electromagnetic wave of frequency lying between Y and v + dv. We have called these photons the S2,v photons. The surface that bounds the volume Y is A. The photon balance is (Figure 5)

+

net flux of Q,Y photons leaving the volume Vacross the surface A

In symbolic form,

(33)

Thus, t o evaluate the LVREA we must know the spectral specific intensity a t each point inside the reactor. Its value can be obtained from the photon transport equation.

time rate of change of Q,v photons in the volume 5’

Figure 6. Representation of intervening phenomena in radiation transport. (1) Incident intensity along s with direction R and frequency Y , (2) emission of radiation in the direction R and frequency v (gain of radiation in the direction R and frequency v ) , (3) intensity in a representative, arbitrary direction R’ and frequency v’ to be scattered in the direction R and frequency v (gain of radiation in the direction R and frequency v), (4) absorbed intensity (loss of radiation in the direction R and frequency v ) , ( 5 ) scattered intensity in a representative, arbitrary directionR” and frequency v‘ out of the direction R and frequency v (loss of radiation in the direction R and frequency Y ) , and (6) emerging intensity along s in the direction R and frequency v (after losses by absorption and out-scattering and gains by emission and in-scattering). [Adapted from Martin et al. (1993).1

In eq 35, n,,, is the number of photons per unit volume, unit solid angle of propagation, and unit frequency interval; n;,, represents the rate of S2,v photon absorption and n i , ythe rate of Q,Y photon emission, both per unit time, unit volume, unit solid angle, and unit frequency interval. ngl: is the rate of photon gain due to in-scattering along the direction Q and nr:t is the rate of photon loss due t o out-scattering hom the direction Q, both per unit time, unit volume, unit solid angle, and unit frequency interval. In the right-hand side emission is important for processes at high temperatures or when fluorescence or phosphorescence phenomena are very significant. In homogeneous systems, in- and out-scattering can be neglected; however they are important in solid-fluid or gas-liquid systems, for example in semiconductor photocatalyzed reactions. In eq 35 we can do the following operations: Since V is fixed we can interchange differentiation and integration in the first integral, also we can apply the divergence theorem in the second integral, and finally we can put all the terms under the same volume integral sign. Since the limits of integration ( V) are arbitrary, after multiplying by the energy of a S2,v photon and recalling eq 28, for a direction S2 and a frequency v, we have

Equation 36 is the general form of the radiation conservation equation for S2,v photons.

Ind. Eng. Chem. Res., Vol. 34, No. 7,1995 2166 Usually the first term can be neglected (the factor l/c makes it always very small); i.e., at a given time the radiation field reaches its steady state almost instantaneously. However In,,, will change with time if the boundary condition associated with eq 36 is time dependent (for example, in a photoreactor having a time dependent radiation source emission) or if the state variables which appear in the constitutive equations for any one of the different processes WQ,v, Wen,+,, W;f;l, and Ws-out change with time. In eq 36 two source terms and two sink terms have been written. For each one of them we need constitutive equations. 1. Constitutive Relationships for the Photon Transport Equation. Absorption, emission, and scattering are phenomena that need some kind of modeling (Figure 5). As usual, constitutive equations for the material behavior provide this representation. Absorption and emission are rather well-known phenomena in the area of homogeneous‘photochemistry. A medium may be regarded as optically homogeneous provided that the linear dimensions of the inhomogeneities are sufficiently small as compared with the wavelength of the radiation. However, when radiation travels through a medium that has inhomogeneities such as small particles, radiation rays will be scattered in all directions. For an exact treatment of scattering, the state of polarization of the radiation field is also important. In general, linear isotropic constitutive equations are used t o characterize absorption and out-scattering. For absorption one can write (37)

Equation 37 gives the absorption of the incident radiation specific intensity by the matter per unit time, per unit volume, per unit solid angle of incidence and per unit frequency. In eq 37 Kv(x,t)is the linear or volumetric absorption coefjricientthat represents the fraction of the incident radiation that is absorbed by the matter per unit length along the path of the beam. This constitutive equation, when inserted into the radiation photon transport equation for a purely absorbing medium, gives rise t o what is generally known as the “Bouguer-Lambert law” for radiation absorption in homogeneous media. Out-scattering is a process by which radiation transported in a given solid angle d P about the direction S2 is taken out of such a direction of flight by matter and sent into all directions in space. For example, one of such directions in space could be another solid angle of propagation dP” about the direction W (Figure 5). In fact, out-scattering can produce also a modification in the frequency of the scattered radiation. Be it a change in direction or a change in frequency, some energy is taken out of the set formed by the S2,v photons. The out-scattering is represented by

Equation 38 gives the scattering of the incident radiation specific intensity by matter in all directions in space and in other frequencies, per unit time, per unit volume, per unit solid angle, and per unit frequency. In eq 38, av(x,t)is the linear or volumetric scattering coefficient. The scattering coefficient represents the fraction of the incident radiation that is scattered by the matter in all

directions and in other frequencies, per unit length of the path of the beam. The scattering coefficient does not provide any information as to the directional and frequency distributions of the scattered radiation. Both distributions can be described by the phase function that is proportional t o the probability that incident radiation arriving from a solid angle dP’about the direction Q’ and having a frequency v’ will be scattered into a solid angle dS2 about the direction S2 with a frequency v. As far as the photon transport equation for Q,v photons is concerned, the phase function is not required for modeling out-scattering. More details about the phase function will be given when treating in-scattering. Emission varies very much from one process to another. Most bodies have a spontaneous emission that depends, a t least, on their temperature; the process is significant a t high temperatures (a radiating body, a flame, etc.). In other cases upon excitation by other external fields (an arc light produced by an electrical discharge between two electrodes, fluorescence generated by a radiation field, etc.) bodies can emit radiation. In general, one can write

For example, the spontaneous emission by a body is represented by

where lv,b(X,t) is given by some form of the Planck equation for blackbody emission. In writing eq 40 local thermodynamic equilibrium has been assumed and that the emission process follows the Kirchoff equation. Particular models employed in photochemical reactors for lamp emission will be treated in detail below (section

M). In-scattering is responsible for most of the complications that arise when scattering of radiation is an important phenomenon. Part of the radiation that is scattered in space may be incorporated t o the stream of S2,v photons according t o the scattering distribution function (the phase function). This process may occur from all directions in space and for all frequencies. Thus, (Qv’), (W,v), and (Q’,v’) photons may be incorporated to the family of (S2,v) photons. The contribution must be computed for all directions and all frequencies. When radiation arrives at the volume element from all directions and all frequencies, this integration yields

wit; = 4n 1 fR’=4n f -

v‘=O

av(x,t)p(v’-v,Q’--n)

In.,,.(x,t)dv‘ dS2’ (41) where p is the phase function. Usually the phase function is normalized according to

The above equation is valid for incoherent scattering. The scattering is incoherent or anelastic when the frequency of the scattered radiation is different from that corresponding to the incident radiation; i.e., the radiation beam changes its energy. Otherwise the scattering is elastic or coherent. For coherent scattering


(43)

substitute the different constitutive relationships. M e r defining a directional coordinate s along the ray path,

with (44)

The problem may be further simplified if the assumption is made that the scattering is single. In multiple scattering, energy scattered by one particle can hit other particles and be scattered additional times. If L is the characteristic dimension of the system, scattering may be considered single if ad. I 0.1 (Siegel and Howell, 1992). It can be seen that the product ad. is the optical thickness for scattering. Another limiting case of simplicity is the case of isotropic scattering; in such a limiting casep = 1. Isotropic scattering requires, among other requirements, that at least the scattering material be homogeneous and isotropic, and that the surrounding medium be also isotropic. More details on phase functions can be found in the classical references of van de Hulst (1957), Ozisik (19731, and Siegel and Howell (1992). In general the absorption coefficient is a function of the state variables, frequency, and concentration of the absorbing species. The state variables such as P and T influence the electrical resistivity, the magnetic permeability, and the electromagnetic permittivity, all of which define the conductivity of a given material. Thus, one can write K,(X,t)

= K,(P,T,Ci)

(45)

On the other hand, the scattering coefficient also depends on the physical cross section of the material in a very complicated way. This dependence has a wide range of variations. For this reason different theories have been proposed t o interpret this phenomenon. A characteristic, dimensionless length is usually used to define different types of scattering: x = ndJA,. When x < [0.6/(refraction index)] or A, >> d,, Rayleigh theory applies; in this case scattering depends on the square of the particle volume. When [0.6/(refraction index)] < x < 5, or 1, PZ d,, Mie theory should be applied; for scattering, it renders a complicate relationship with the particle diameter. Finally for x > 5 or A, 4. Other studies have indicated that independent scattering exists when the particle volume fraction cfv) is smaller than 0.006 and that for fv > 0.006 it is necessary that NAPL 0.5. Independent scattering is a condition for using linear relationships between the scattering coefficient and the particle concentration (Siegel and Howell, 1992). Perhaps one of the most important conclusions that can be drawn from this equation is that in heterogeneous reacting systems classical forms of analyzing the light distribution inside the photochemical cell (i.e., the Lambert-Beer equation) would be incorrect and, very likely, useless. This conclusion is particularly important when the effect of the incident radiation on solid-fluid photocatalyticreaction rates is investigated. The second conclusion of significance is that, with only a very few exceptions, radiation transport is a three-dimensional phenomenon, and one-dimensional models cannot be used with confidence always.

VII. Radiative Transfer in Participating, Reactive, and Homogeneous Media (Single, Fluid Phase Photoreactors) Let us consider a homogeneous and fluid medium where a photochemical reaction occurs. Radiation arrives at any point in the reaction space from a lamp that emits in all directions (spherical emission in space). Lamps are usually located separated from the reacting section. They may be part of a continuos flow, annular reactor (see Figure 1)or they may be immersed together with an immersion well (jacket) in a well-mixed tank reactor. Obviously, many other lamp-reactor arrange-


Figure 7. The radiation balance in a homogeneous system.

I

Reactor

Figure 6. Application of radiation balance. [Reproduced from Cassano et al. (199413) with permission. Copyright 1994 Research Trends.]

ments may be used as will be illustrated in the section dealing with applications. Clearly, the reacting system can be divided into three sections as far as the radiation field is concerned (Figure 6): (1)the lamp space, for the moment assumed to be a volume of emission, (2) the cooling space which for most applications is diactinic, and (3) the reaction space. Let us look a t a radiation beam going from the lamp to a point inside the reactor such as is illustrated in the figure. From ~ 2 ( O , 4 )to el(O,#)there is only emission of radiation (it will be assumed that emitters do not absorb radiation and that scattering inside the lamp is negligible). From el(O,#)to oi(O,4)it will be assumed that the medium is diactinic (no absorption, no scattering, and no emission). From ei(O,4) to eI(O,4) there is absorption and reaction (assumptions will be made that in the reacting chamber there is no emission of radiation and that, for this case, the medium is homogeneous). Inside the reactor and for an arbitrary direction (e,+) one can write the radiative transport equation. In e = ei we will need a boundary condition. Since specific intensities in the diactinic section (section 2) are preserved, the specific intensity at e = el will be equal to that entering the reactor at e = ei. This means that, for a given direction of propagation, the emerging intensity from the lamp will provide the boundary condition for the radiative transfer equation inside the reactor; i.e., the boundary condition for the radiative transfer equation inside the reactor is the intensity at the point located in e = el, and is the point on the lamp seen in the Q direction by the incidence point I. "he modeling of the radiation emission by the lamp will provide the required information. This topic will be treated in detail in the section concerning the Emission Models for Tubular Photochemical Lamps (section E). Let us look at the radiation balance inside the reactor. As indicated above, under normal conditions radiation arrives a t a point I, located at x in the reactor following a light path that is characterized by the directional coordinate s, where s = s(x,O,#)(Figures 7 and 8). Radiation will be attenuated by absorption along its trajectory. Since the medium is truly homogeneous, no scattering is possible. Let us assume, additionally, that operating conditions are such that no emission of radiation exists besides that corresponding to the lamp. Radiation from the lamp arrives a t the reactor wall at a point where s = SR. At the wall the ray may be transmitted, but also partially absorbed, reflected, and

Figure 8. Coordinate system.

refracted. Absorption and reflection may be readily accounted for through the use of a transmission coefficient (or separately, with absorption and reflection coefficients); refraction is a phenomenon much more difficult to model. Besides, in a first approximation refraction can be considered a second-order effect and, for our purposes, its consequences may be neglected. After traveling a length (s - SR) the ray reaches the point under consideration (I) a t x. Let us remark that, in the general case, the physical and chemical characteristics of the reacting space may be different from one point to another and, moreover, these properties could also be a function of time. Hence, the specific spectral intensity will have a nonuniform spatial distribution and, even if the emission from the lamp is steady, it may also have a dependence on time; i.e., I,,ds,t) = IV(x,O,4,t). The radiative transfer equation may be written as

Inside the reactor there is no emission Ij:(s,t) = 01; besides, the medium has been assumed homogeneous [a,(s,t)= 01. Thus, eq 51 reduces to

with a boundary condition at the point of entrance given by

~,(~,,n,t) = t ( n , t )= @ e , w

(53)

Equations 52 and 53 may be formally integrated from the point of entrance at the reactor wall (s = SR) to the


point under consideration (s = s) t o give

From eqs 30 and 33, the LVREA results (55)

e;(x,t) = ~ , ( x , t )Iv(x,8,4,t) h~ dQ

In the integral of eq 55 the region of integration for the incident radiation may be divided into two parts: (i) the first, corresponding to the solid angle delimited by the lamp boundaries (Q,), that is the solid angle that transports radiative energy coming from the body of the lamp, and (ii) the solid angle corresponding t o the remaining of the surrounding space. Then,

Obviously, no radiation comes from the space defined by the solid angle 4n - 52, and the only nonzero integral is the first. The final result can be written in terms of a spherical coordinate system with origin a t the point of incidence I (Figure 8); then,

Strictly speaking, from a conceptual viewpoint, eq 57 is the same equation that is used in photochemical kinetics to calculate the absorbed incident radiation (erroneously called “absorbed intensity” and perhaps also misleadingly designated as I*). The difference resides in its more general validity because no particular restrictions have been imposed in its derivation. In fact it is valid for (i) monochromatic radiation, but an integration over all significant frequencies turns it equally useful for polychromatic sources of energy; (ii) noncollimated radiation (i.e., radiation arriving from all directions in space from a radiation source that emits in a spherical manner), but its unidimensional form can be obtained with a straightforward procedure; (iii) any form of absorption, since nothing has been said about the way in which the absorption coefficient [~,(s,t)lis related to the concentration of the absorbing species (for dilute concentrations the so-called Lambert-Beer equation could be used); and (iv) point (local) values of the LVREA (but if the global value is desired, it can be obtained with a simple volume-averaging procedure). However, it must be stressed that eq 57 has a limitation of practical value. It was not meant for and it does not represent situations where inhomogeneities are present. Inclusion of scattering for heterogeneous media (for example, photocatalytic ones) means considering an additional, different class of phenomenon, and in general, a simple exponential decay for attenuation will not be valid.

MIL Radiative Transfer in Participating, Reactive, and Heterogeneous Media (Multiple Two important observations must be made in connection with eq 57: 1. After writing the equation in terms of the spherical coordinate system, the solid angle corresponding to the space defined by the lamp boundaries (52,) originated a first integration in e(#) and a second one in 4. These integrals have limits of integration that are complex functions of the system geometry. The methodology for obtaining these limits will be illustrated in a specific section of this paper following an approach proposed by Irazoqui et al. (1973) (section X). If reflectors are part of the illuminating system, these complexities are even more severe as will be shown in the section of this paper dealing with reflecting surfaces; the general methodology was derived from the original work of Cerdsl et al. (1973, 1977) (section XVII). 2. In eqs 54 and 57 the boundary condition has taken the form of c(O,$,t). As indicated before, this value must be provided by a lamp emission model. At this point it suffices to say that lamp characteristics system geometry

]

(58)

Equation 57 provides, in mathematical form, all the elements needed t o compute the LVREA in homogeneous media. With an emission model for the lamp and the proper integration limits for the lamp radiation contributions to an arbitrary point of incidence (I), the local value of the radiation energy absorbed per unit time and unit reaction volume can be obtained. This is the value required to formulate any homogeneous photochemical reaction kinetics with monochromatic radiation. Extension t o the more general and practical case of employing polychromatic radiation requires an integration for all intervening wavelengths as will be shown below (section XVI).

Phase Photoreactors) When the reacting medium is heterogeneous, scattering effects become important. Let us assume that changes in the direction of flight are the main scattering effects; i.e., the scattering is coherent. To obtain an equation valid for heterogeneous media, let us insist that a very important additional assumption is required. To use the working form of the photon transport equation, one must assume that in each material point in space the volumetric absorption coefficient and the volumetric scattering coefficient are welldefined properties of position and time. In other words, given a position s and a time t in our system, one is always required to define properties (such as density and/or concentrations and/or temperature, for example) of the material point occupying such a position in space and time. This is the same as saying that we are forced to treat our system as pseudo-homogeneous. Then, the system properties (the optical ones included) are assigned to the material point and not to anyone of the particular phases that may be present. Under this condition, the radiative transport equation can be applied in the pseudo-continuous domain to obtain

with

Equations 59 and 60 may be formally integrated from the point of entrance at the reactor wall (s = SR) to the

point under consideration (s = s, Figure 9) to give eq 61 in Table 6. It can be readily seen that the first term in eq 61 gives the extinction of the incoming radiation from the lamp and the second one gives the extinction of the radiation incorporated into the direction 2 ! by inscattering. Thus, clearly, for the direction under consideration (a),in-scattering acts as a source term. In Figure 9 one of the many points of in- and out-scattering along the ray trajectory has been drawn. Equation 61 may be written in terms of a spherical coordinate system located at the point of incidence (recall Figure 8) as indicated by eq 62 in Table 6. Finally, the value of the LVREA is given by eq 63 in Table 6. Equation 63 can be written in a more explicit form as indicated by eq 64 in the same table. Equation 64 gives a clear indication of the difficulties associated with the characterization of the absorbed incident radiation in a heterogeneous photochemical reactor, for example in a photocatalytic reaction using solid semiconductors. In eq 64 K , , ( s , ~ ) , a,,(s,t), and the phase function p(O’,@’+O,@)must be provided by independent theoretical considerations and/or experimental measurements. Although it is impossible at the present time to use theory for predicting accurate values of the absorption and scattering coefficients, some attempts have been made to formulate phase functions for very particular conditions. Examples of them are the cases of Rayleigh scattering for particles of very small size (dp < 0.030.06 pm for near-W radiation), Mie theory for particles of any size, and geometrical optics (specular and diffise reflectance models) for particles of relatively large size (dp =- 0.5 pm for near-UV radiation). The interested reader can find a concise treatment of the subject in the book by Siege1 and Howell (1992). To complete the formulation of e: one needs to know (1) the limits of integration e(@)and 4 for the first integral of the right-hand side of eq 64 and (2) the boundary condition I,(s~,n,t)= c(O,@,t). The boundary condition depends on the type of emission of the used lamp and the geometrical properties of the lamp-reactor system. Conversely, the limits of integration only depend on the geometry of the reacting system. Both aspects will be treated in the sections that follow.

IX.Emission Models for Tubular Photochemical Lamps

As indicated before, the boundary condition for the radiative transfer equation inside the reactor should be provided by the modeling of the lamp emission. Up to the present time, in photoreactor engineering applications tubular lamps have been used almost exclusively. Two main types of lamps can be modeled. There are lamps that produce an arc that emits radiation by itself, and consequently, photons come directly from such an arc. Emission is made by the whole lamp volume. For example, this is the case of mercury arc low-, medium-, and high-pressure tubular lamps. We speak in these cases of voluminal emission. There are other types of lamps in which the discharged arc between electrodes induces an emission produced by some particular substance that has been coated on the lamp surface. For example, this is the case of any type of fluorescent tubes, some black light lamps, actinic lamps, etc. In these cases the actual emission is produced by the lamp envelope; i.e., by the surface of the lamp. We speak in these cases of superficial emission. Voluminal emission may be safely modeled as an isotropic emission; in this case the specific intensity associated with each bundle of radiation originated in some element of volume of the lamp is independent of direction, and the associated emitted energy (per unit time and unit area) is also isotropic (Figure 10). On the other hand, it seems that superficial emission can be better modeled by a diffise type of emission that is also known as one that follows Lambert’s “cosine law” of emission; in this case the emitted intensity is independent of direction but the emitted energy depends on the surface orientation and follows the “cosine law” equation (Figure 10). Both models for extended sources (lamps represented as three-dimensional bodies) will be presented here. The following assumptions are made (Irazoqui et al., 1973): 1. The emitters of the radiation source are uniformly distributed over the region of emission (a volume or a surface). 2. In terms of specific intensities each elementary extension of emission has an isotropic emission but the outgoing radiation energy is (i) isotropic when the


I

-s

:

Reactor

S

representative center of scattering

Figure 9. The radiation balance in a heterogeneous system. Figure 11. The extended, superficial, diffuse emission (E-SDE) source model. [Adapted from Irazoqui et al. (19731.1

TUBULAR LAMPS

c(0,4>= Iv(x,@,4)le=ei = I,(X,@,4>le=eeYR,v(n) (66)

where YR,JQ) is the reactor wall transmission coefficient. Since emission is uniform in space and isotropic in directions,

Intensity and Energy flux

VOLUM INAL (ISOTROPIC 1

Now we must relate the value of the specific intensity of emission t o the emission power of the lamp. From the definition of the spectral specific intensity,

SUP€ RFIC IAL (DIFFUSE )

Figure 10. Voluminal and superficial emission. [Reproduced from Cassano et al. (199413) with permission. Copyright 1994 Research Trends.]

dp,,, = I",

aQ d ~ cos , 0,

(68)

From which emitting element is a volume or (ii) diffuse (affected by the surface orientation) when the element is a surface. 3. Any emission element of the lamp emits per unit time, and for a given frequency, an amount of energy proportional to its extension and independent of its position inside the lamp volume or on the lamp surface. 4. When emission is voluminal, each of the differential volumes of emission is transparent to the emission of its surroundings (very likely, an approximation). 5 . The lamp is a perfect cylinder bounded by mathematical surfaces with zero thickness. Hence, any bundle of radiation coming from inside does not change its intensity or direction when it crosses this boundary (again, an approximation). 6. The lamp is long enough; consequently, neglecting end effects, the emission produced by the lamp along its central axis is uniform. This assumption does not impose uniformity on the radiation field generated along the direction of the central axis. 7. Emission from the lamp is at steady state. 1. Three-DimensionalSource with Superficial, Diffuse Emission. The E-SDE Source Model (Figure 11). From assumption 7, emission is not time dependent. From the lamp surface (s = ss) to the reactor wall (s = SR) there is no emission, no scattering, and no absorption (the medium has been assumed t o be diactinic); then,

(65) It follows that

q=

PV,S

lQ,cos 0, d Q j , dA,

-

pv,s

(69)

~ ~ R L L L

According t o eqs 66, 67, and 69 the boundary condition for eqs 57 and 64 when a lamp with superficial emission is used is given by

(70) 2. Three-Dimensional Source with Voluminal, Isotropic Emission. The E-VIE Source Model (Figure 12). Since emission is produced by a volume, the radiative transfer equation can be applied inside the lamp. There is no absorption (assumption 4) and no scattering. The resulting equation is

From assumptions 2, 3, and 6, emission is isotropic and uniform. Hence, at steady state (72)

Along the direction Q, at s = 0, there is no entrance of radiation; this situation provides the required boundary condition for eq 71: s =0

IV(O,Q)= 0

(73)

Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 2171 Integrating from s = 0 to s = ss and using the following change of coordinates (recall Figures 8 and 121,

as=-@

(74)

e=e2 @=el

s=O s=s,

(75)

(76)

one gets

with

[AeS(x,8,4>l = e2(x,8,4)- el(x,0,4)

(78)

It must be remarked that, as it should have been expected, Aes is a function of the position x inside the

Figure 12. The extended, voluminal, isotropic emission (E-VIE) source model. [Adapted from Irazoqui et al. (1973)and CerdP et al. (1977).1

reactor and the direction of the incoming radiation given by the spherical coordinates (e, 4). Once more, from s = ss to s = SR there is no emission, no scattering, and no absorption (the medium has been assumed t o be diactinic); from assumption 5 there is no refraction or reflection a t the lamp boundaries, then

Consequently,

C(e,#) = I,,(sR,Q)= I,(s,,Q>y R ~ s - 2 )

(80)

One can note that eq 66 was obtained with a similar reasoning. Finally, the boundary condition is

C(e,#)=j:[Ae,(x,8,4)1Y,,JQ)

(81)

As indicated before, reflection and absorption at the reactor wall were accounted for by means of the wall transmission coefficient. Once more, the value of j : must be related to the By definition,j: is the energy lamp output power PV,,, emitted per unit volume, unit solid angle of emission, and unit time; then

Figure 13. Continuous flow,annular photoreador. [Adaptedfrom Irazoqui et al. (1973)and De Bernardez and Cassano (19851.1

Using the nomenclature indicated in Figures 12 and 13, the equation of the boundary surface of the radiation source (a cylinder) in spherical coordinates is written as follows:

e2sin28 - 2e(sin 8 cos Q,)r+ (r2- RP) = 0 Since j : corresponds to an isotropic and uniform emission, D

The two solutions of this quadratic equation are precisely the values of e; i.e., they are the intersections of the e coordinate with the front and rear parts of the lamp at any value of 8 and 4: e1,2 =

From eqs 78 and 81 we must know the values of e2(x,6,#)and el(s,6,q5). In order t o know these values, one must obtain explicit expressions for the independent variable e, a t the positions indicated by eqs 75 and 76. To illustrate the procedure, the case of an annular reactor will be analyzed (Figure 13). Let us consider a point located in an arbitrary position I, having coordinates x (r, ,6,z). Let us look at an arbitrary direction

(e,#).

(85)

+

r COS Q, f (r2cos2 Q, - r2 RL2)1’2 (86) sin 8

Finally, the following value for Aes is obtained: 42s=

+

2[r2(cos24 - 1) R,211’2 sin 6

(87)

Equations 81, 84, and 87 provide the boundary condition for eqs 57 and 64 when a lamp with voluminal emission is employed:

2172 Ind. Eng. Chem. Res., Vol. 34,No. 7,1995

X Limits of Integration for the LVREA When Using Tubular Lamps When a lamp with superficial emission is used, according to eq 70,a constant value must be incorporated as a boundary condition to eqs 57 or 64. Conversely, when lamps with voluminal emission are used, according to eq 88, the boundary condition introduces a function of x, 8, and 4 into eqs 57 and 64. For both cases, the limits of integration for the independent variables 8 and 4 will be obtained. 1. Limits for the Independent Variable 8 (Figures 11, 12, and 14). Limiting rays coming from the lamp and reaching the generic point (r,j3,z ) must satisfy two conditions. (i) If the ray limits the value of 8, its equation (a straight line in space) must have a common solution with the equation of the circumference that defines the opaque zone of the upper and lower parts of the lamp; however, for any plane at constant (in r-2) there are two values of 8 that satisfy this condition for both the upper and lower boundaries. (ii) To eliminate this ambiguity a second restriction on 8 must be imposed: one must choose the angle corresponding to the intersection of the ray with that portion of the circumference that, limited by the two generatrix lines corresponding to the limiting angles of 4, is closer t o the generic point (r,p, z), as indicated in Figure 14. From Figures 13 and 14,

(LL - 2 ) = el COS -2

el

= el cos e,

(89) (90)

and with eq 86,

Figure 14. Limits of integration for the source. [Adapted from Irazoqui et al. (1973).1

The limits of integration for the case in which reflecting surfaces are present in the system (for example, elliptical or parabolic reflectors) require a more elaborated procedure. Complications are only due to the intricate trigonometric and algebraic manipulations that are needed to account for the reflection by a curved surface, but no additional concepts are required. The procedure will be analyzed in a specific section of this paper (section M I ) .

XI. Working Equations for Calculating the LVREA

2. Limits for the Independent Variable 9 (Figure 14). The limiting rays in the 4 direction, for any value of the angle 8, must be tangent to the lamp boundary at points located on the two generatrix lines of the cylinder (Figure 14). These values can be obtained by imposing a restriction in the values of el and ~ 2 At . the limiting points, both intersections of the e coordinate with the lamp boundary must coincide; i.e., in a projection of the lamp on the x-y plane, the limiting rays are tangent t o the directrix circumference of the lamp. This means that the condition in space is

el = e2

(93)

From eq 86 the following equation is obtained: r2 cos2 4 = r2 - ~

, 2

(94)

and since 4 can only take on values in the first and fourth (negative value) quadrants, we have

All the required equations for calculating the photochemical energy absorbed at one point inside the reactor have been obtained. Let us summarize the results. 1. Homogeneous Systems. 1.1. Lamps with SuperficialEmission. The boundary condition is incorporated into eq 57 t o give eq 96 (see Table 7). This equation must be used with the limiting angles given by eqs 91,92, and 95. The lamp manufacturer should provide the value of Pv,s. The dimensions of the lamp as well as those of the reactor are known. The absorption coefficient can be obtained from an independent measurement and by using some relationship for its dependence on concentration. Hence, a t any point inside the reactor and a t any time, the value of ea can be calculated. This is a typical result that shows why iterative methods are always required to design a photoreactor: preestablished reactor dimensions are required to calculate the LVREA. 1.2. Lamps with Voluminal Emission. When the boundary condition is incorporated into eq 57,eq 97 is obtained (see Table 7). As in the case of eq 96 the limiting angles are obtained from eqs 91,92, and 95. All the required information is k n o w n or it can be obtained from independent measurements. Hence the value of the LVREA, a t any

point inside the reactor and at any time can be calculated a priori. No experimentally adjusted parameters are needed. 2. Heterogeneous Systems. 2.1. Lamps with Superficial Emission. For this kind of system eq 98 is obtained (see Table 7). 2.2. Lamps with Voluminal Emission. In a similar way, for this model we get eq 99 (see Table 7).

XII. The Absorption Coefficient The specific intensity (Iv) does not change with distance only when the propagating medium is transparent and there is no emission of radiation. When absorption occurs, I , changes as is established by the Bouguer-Lambert equation for homogeneous media; however, if other phenomena are present, for example, scattering, the quoted equation is not sufficient t o describe the involved phenomena. Consequently, in the evaluation of the absorption coefficient it is necessary to make a distinction between homogeneous and heterogeneous media. The latter are encountered when one attempts to model photochemical systems in which gas bubbles are dispersed in a liquid, or solid particles are incorporated into a gas or liquid phase as in photocatalytic applications. The distinction arises from the fact that the presence of heterogeneities introduces unavoidable light dispersion (in- and out-scattering) in the radiation field. For this reason it becomes important to emphasize that the Bouguer-Lambert equation is a constitutive formulation valid only for a one-directional phenomenon: the absorption of radiation along the direction of propagation. Recalling eq 52 absorption is a process that involves only the S2 direction. When phenomena other than pure absorption are present, the propagation of a bundle of rays along a given direction becomes a process that requires a more elaborate representation that cannot avoid the simultaneous consideration of all other directions of radiation propagation in the three-dimensional space. In eq 59, inscattering considers all other directions Q’ from space. If this is the case, the use of a “Lambertian” form for representing absorption and scattering may introduce significant errors. 1. Homogeneous Media. In a homogeneous medium, the radiation intensity along a given direction of propagation changes only because of the absorption process in the material volume. As mentioned before,

this variation in the specific intensity along the ray trajectory may be expressed by a constitutive equation that involves an absorption coefficient K v j . This coefficient is a function of the frequency of radiation v and the state variables such as temperature, composition, and, in some cases, pressure. The exact form of the absorption coefficient function may be determined from the microscopic theory of matter by studying the absorption of photons by atoms or molecules exposed to a radiation field. Macroscopically, the volumetric absorption coefficient (also known as the Naperian absorption coefficient) is considered t o be a linear function of the concentration of the absorbing species: (100) When Cj is a molar concentration, a + j is the molar Naperian absorptivity of the radiation-absorbing species j . If more than one species ( K = 1, 2, ..., m)participate in the radiation absorption process, eq 100 may be written as k=m

k=m

k=l

k=l

Both eqs 100 and 101 are strictly valid only for dilute solutions; when they are used with the constitutive equation for absorption (eq 52), the result is usually known as the “Lambert-Beer law”. In a reacting medium eq 101 must be used with great care. With the exception of a photosensitized or photocatalyzed reaction (where the significant absorbing species is not consumed), the value of the absorption coefficient becomes a function of the extent of reaction. However, in eq 101we are including all possible species ( K ) that absorb radiation, and this is a point that deserves a detailed treatment. Three types of species may be considered: (i) a radiation-absorbing reactant, (ii)a radiation-absorbing product, and (iii) a radiationabsorbing inert. Referring to eq 57, one can observe that the absorption coeficient K~ intervenes first in a linear form (say K:) and in a second instance as part of an integral in the exponential function (say K : ~ ) . If we recall eqs 23 and 32, the linear dependence represents the direct kinetic effect of the radiation absorption process; consequently, this coefficient (K:) must include only those species that take explicit

2174 Ind. Eng. Chem. Res., Vol. 34, No. 7,1995

participation in the initiation rate and exclude the radiation absorption produced by inerts and/or reaction products. [Products may have their own participation in the reaction kinetics in two possible ways: (i) when the reaction is reversible and (ii) when upon radiation absorption the product undergoes a secondary reaction; however, these are different phenomena that must be analyzed in a different context.] As explained before, the most common cases are (i) a photosensitized reaction, where since the concentration of the sensitizer does not change with the progress of the reaction, K: remains constant; and (ii) reaction beginning with absorption by one of the reactant species that is consumed during the reaction, and then K: becomes a function of the evolution of this reactant concentration during the course of the reaction. Conversely, K F participates ~ in the evaluation of the incident radiation G,, and here the integral in the exponential function represents the attenuation of the radiation beam by the absorption process along the beam trajectory. It includes the contribution of all types of absorbing species; since some of them may be reactants and/or products, also ~y~ will usually be a function of the reaction extent. Then, in practice, eq 57 represents a very simple case [only one of the species (a reactant) absorbs] and, consequently, only one value of K~ has been used; i.e., inerts and/or products are transparent t o radiation. Problems of a more complex nature have been solved by Alfano and Cassano (1988b1, Cabrera et’al. (1991c), and Tymoschuk et al. (199313). 2. Heterogeneous Media. Since homogeneous systems have been the main objective of most of the research performed so far in photochemical reactor analysis, the information available in the literature about modeling attenuation in heterogeneous media is particularly scarce. A critical review about this subject can be found in Alfano et al. (1986b). The problem can be approached in two different ways: (i) by modeling the process taking into account the scattering of radiation produced by the heterogeneities of the medium and (ii) by using the existing formulation on homogeneous media with an effective absorption coefficient. The first approach involves the rigorous solution of the complete radiative transport equation and the use of much more information than a single absorption coefficient. In fact, it is additionally required to know the scattering coefficient and the phase function for the distribution of the scattered radiation. Using this strategy, precise modeling of a practical, heterogeneous reactor has not been experimentally validated yet. We will deal with this subject in a specific section of this paper (section XX). Using the second approach, at present, several correlations have been proposed t o evaluate the effective attenuation coefficient in gas-liquid dispersions. Among them, we can mention particularly two: 1. The first one, proposed by Otake et al. (19811, is a simple empirical expression that accounts for the absorption effects produced by the liquid phase and the reflection and refraction effects provoked by the gaseous phase. Considering that the latter are proportional to the specific surface area (Av), ~ ~ , may ~ f be f represented by the following correlation:

where K” is the attenuation coefficient of the liquid phase, EG is the holdup of the dispersed phase, and k is an empirical coefficient depending on the optical proper-

ties of the system. Otake et al. plotted the experimental information obtained by them and other authors, finding that the values can be correlated well by using k = 0.125 for K ” , ~ RK”, , and A, in the same units (cm-l). 2. The other correlation was proposed by Yokota et al. (19811, in which the effective attenuation coefficient becomes a function of the attenuation coefficient in the liquid phase, the bubble diameter, and the gas holdup. The expression is K,,,ff

= KY{ 1

+ [(3.6/d,)o’66]€~}

(103)

where db is the bubble diameter in mm. Both correlations may be used in the case of K” smaller than 40 m-l. When K, is greater than 40 m-l, the dispersing effect of radiation due to the presence of bubbles may be normally neglected. These correlations have been successfully used by Alfano and Cassano (198813). Neither similar nor reliable correlation has been found for solid-fluid processes yet (or for gassolid, liquid-solid, or gas-liquid-solid systems).

XIII. The Coupling Radiation Attenuation-Reaction Extent

As can be seen from eqs 96-99, the LVREA generally results a function of the spatial variables, of the concentration of the absorbing species, and of other physicochemical parameters (in heterogeneous media, particle size and particle concentration are, among others, additional variables). Let us look at the case in which the radiation-absorbing species may undergo changes in concentration. If a species (i) which is being considered in the mass balance is a radiation-absorbing species (j = i in eq 1001, a coupling between the LVREA and the corresponding mass conservation equation will take place. However, there exist exceptions represented by photosensitized reactions. Here, the absorbing species is activated, interacts with another reactant that is consumed and, after a cycle, is restored. Its original concentration does not change. If the other reacting species or products do not absorb light, the radiation field is independent of the extent of the reaction. We can say that there is no coupling between the attenuation of radiation (caused by absorption) and the extent of the reaction. Heterogeneous photocatalytic reactions also follow this reaction pattern. When this is not the case, the absorbing species reacts and its concentration changes with the extent of the reaction. Then, the attenuation effects in the radiation field depend on the distribution of concentrations in time and/or space. Two different situations may occur: (i) a very well stirred reactor, where the attenuation process is, at the most, a function of time and (ii) a reactor with space dependent concentrations. In case ii the most difficult problem produced by the coupling attenuation of radiation-extent of the reaction arises: spatial and sometimes also temporal changes in the concentration of the radiation-absorbing species produce changes in the attenuation of radiation within the reaction space. This coupling is very complicated as shown in Figure 15. The LVREA is not only a point function of the reactor position, but also a functional of all the optical properties of the space through which every ray of radiation reaching point I has traveled:


Figure 15. Schematic representation of the coupling “attenuation of radiation-extent of reaction” for the evaluation of the LVREA. [Reproduced from Cassano et al. (1994a) with permission. Copyright 1994 Research Trends.]

In eq 104 E Q(x)indicates that there is a solid angle of irradiation at point x (the angle 52) that defines the space of trajectories along which the radiation intensities arriving at such a point are attenuated by absorption. The position of all the points in space transversed by the different ray trajectories is represented by 6. Thus, C&t> gives the distribution of concentrations of the component j at a time t , within a domain Q(x) dependent on the point x under analysis. In our example the time t is not considered (steady state). The consequence of this coupling is that there exists a very close link between the extent of the reaction in the whole reactor (which controls the radiation absorption by the absorbing species along each of the radiation beam paths) and the radiation absorption rate at a given point. Thus, to know the reaction rate a t this given point of incidence, ( r ,z ) , we must know the value of the radiation field [say the incident radiation G,(r,z)] and the concentration a t such a point; but the value of G,(F,z)depends on all the existing concentrations along the trajectory of every ray reaching point (F,z ) from the whole space of reaction [for example, concentrations affecting 1,(0,4)1,1,(0,4)2,etc. in Figure 151. That is, in order to know the rate of energy absorption a t a given point in the space of reaction, we must know simultaneously the reaction extent at almost every other point of the reactor (or, what is the same, the absorption rate at almost every other point of the reactor). In short, the LVREA is, substantially, an integral of all the radiation attenuations over the space of reaction. In order to solve this cyclic link, an iterative procedure must be superimposed on the numerical technique used to solve the system of integro-partial differential equations. Starting from an initial value of concentration in the whole reactor, one must solve the problem until an essentially “steady state” concentration of reactants is achieved. This is done by sweeping the integration mesh as many times as needed until an acceptable, prescribed minimum error is attained. XIV. Final Equations for the Example: The

Annular Photoreactor We have completed the presentation of all the equations that are needed for modeling the photoreactor. They are momentum balance, eqs 1-4; mass balances,

eqs 5-8; thermal energy balance, eqs 10-15; reaction kinetics, eqs 16-21; and initiation rate, eqs 23, 91, 92, 95, and 96. It can be noticed that the mass and energy balances are represented by two-dimensional models in cylindrical coordinates (r,z); the third dimension is not required due to the azimuthal symmetry of the reactor cylinder. The initiation rate, derived from the radiation transport equation, is represented in a three-dimensional space and in spherical coordinates [3 (or e), e,@]. In both phenomena we have used their natural coordinate systems. These equations represent altogether a system of integro-partial differential equations that have been solved by finite-difference techniques. [Additional numerical techniques are required for heterogeneous systems in order to account for the in-scattering.] However, at each point of the two-dimensional mass and energy integration meshes we need the value of the LVREA that must be calculated from a three-dimensional equation; thus, to compute the incident radiation we must know the concentrations in a three-dimensional space. Making use of the azimuthal symmetry properties, concentrations at other planes different from the one corresponding to the mass balance integration mesh [say the plane (r-z), at /3 = 01 can be obtained by means of a cylindrical projection at constant r and z. Thus, to obtain the existing concentrations for any other desired plane (p = PI, /32, ...,bq,etc.) the known concentrations a t j3 = 0 can be projected on the desired position in space. Normally, these projections will not fall on the regular intervals existing along the ray trajectory integration mesh [the 3 coordinate, from 3 = SR to s = SI, for any (0, 4) direction]. When this is the case, the level rule using two neighborhood concentrations must be used. This approach was used by Romero et al. (1983) and applied in all subsequent work of Cassano and coworkers. An interesting matching of a three-dimensional mass balance in cylindrical coordinates with a three-dimensional radiation balance in spherical coordinates has been presented by Tymoschuk et al. (1993b). XV. Solution of the Example: Predictions from the Annular Reactor Model This section will be devoted t o the presentation of some demonstrative results obtained with the annular reactor model. They provide an indication of the type of answers that can be extracted from a complete mathematical representation of the reacting system. It is convenient to define the following variables: the feed molar ratio:

(105)

the cup-mixing concentration:

At 5 = 1, eq 106 provides the cup-mixing exit concentration. the reactant conversion:

2176 Ind. Eng. Chem. Res., Vol. 34,No. 7, 1995

At 5; = 1, eq 107 provides the reactant exit conversion.

description

the product selectivity: J=4

S,(O = ( C J W x ( C j ( t ) ) , i = 1,2,3,4 (108) J=1

At 5; = 1, eq 108 provides the product exit selectivity. the cup-mixing dimensionless concentration:

(Y(t)> =

f Ur) Wr,t)r d r dB f (Q)r d r 48 (

(109)

the cup-mixing dimensionless temperature:

j Ur) dr,C)rd r 48 f U(Y)YdY dB (

(z(t)> =

Table 8. Reactor and Lamp Characteristics

(110)

(

In order to maintain this section within reasonable bounds, the analysis will be limited to the study of the effects produced by changes in two parameters that affect a distinct characteristic property of any photochemical reactor: the optical thickness [A' = aACs,o(rrr, - =)I. Since the molar absorptivity is a constant, the feed molar ratio (F)and the length of the annular space (no- Q) will be varied. The methane concentration and the inside radius will be kept constant. Many other variables may be changed when looking for the best design; among them we can mention that the type of lamp, its output power (including the spectral distribution of the lamp output), and its length are relevant properties that may affect the reactor performance in an important manner. Other possibilities are the methane initial concentration, the volumetric flow rate (even considering the possibility of operating under turbulent flow conditions with the appropriate changes in the model), the cooling temperature, and the inlet temperature. However, changes in the optical thickness of the reactor permit showing a phenomenon that can be observed exclusively in photochemical reactors: at constant mean residence time, there is a value of the product C&,o(rrr,- rq)that provides the maximum exit conversion of reactanta; beyond it, any increase in either the radiation-absorbing-reactant concentration or the size of the annular space may result in a decrease in the reactor yield. This type of behavior will be discussed in more detail below. The values of the parameters employed to compute the model predictions are summarized in Table 8. In this table, the following information can be found: (i) main dimensions of the reactor, (ii) process operating conditions, and (iii)main dimensions and characteristics of the employed lamp. One important observation has t o be made. It was indicated in the presentation of the different balances employed in the mathematical description of the model that, with the exception of the reaction kinetic constants and the reaction heats, all other properties have been assumed constant (independent of changes in concentration and/or temperature). This restriction can be eliminated with no conceptual difficulties, but at the expense of considerable computer processing time. However, in any real simulation, changes in properties such as density, multicomponent diffisivities, thermal conductivities, etc. must be incorporated; this is par-

reactor dimensions length (LR) inner radius (m) outer radius (-) operating conditions CldCH4 ratio in the feed (F) methane initial mole fraction inlet temperature (To) cooling temperature ( T J pressure average velocity ((u,)) lamp characteristics, Philips 'TL'40W/09N length (LL) radius (RL) nominal power photochemical output power (P)

value

1.60 m 0.05 m 0.055-0.085 m 0.5-9.0 0.03 298 K 288 K 1.013 x 0.35m s-l

Pa

1.20 m 0.019 m 40 W 2.5 10-5 einstein s-1

ticularly true for a case like this example (chlorination of hydrocarbons) where we are dealing with very exothermic reactions and working with high conversions. One example will suffice to illustrate the problem. Changes in the thermal conductivity of the reaction mixture (particularly with temperature) are not negligible; these variations have a significant effect on such a reaction temperature (specially in laminar flow regime) and, consequently, on the reaction rate. In our case, since we are presenting demonstrative results, for simplicity, we decided to use an average value of the mixture thermal conductivity. It was calculated from the values obtained at the minimum and maximum reactor temperatures predicted by the model. Firstly, let us look at concentration and temperature profiles. Figure 16a depicts the dimensionless radial concentration profiles for reactants (broken lines) and products (solid lines) at the reactor middle high position (5; = 0.5); these values are for a constant feed molar ratio (F = 3) and for a constant annular gap (rrr, - r ~ = , 2 cm). T w o different scales for the ordinate have been used: dimensionless reactant concentrations must be read on the left side while those corresponding to products are to be found on the right side. Additionally, since F = 3, to facilitate a better representation, methane concentration has been multiplied by a factor of 4. For reactants (chlorine and methane) one observes (i) a region of maximum conversion (minimum concentration) close to the interior wall of the reactor, (ii) a region of minimum conversion located between both reactor walls, and (iii) a region of intermediate conversion close to the outer reactor wall. This type of behavior is the result of the superposition of two different effects: (i)the attenuation of the radiation field along the radial direction due t o radiation absorption and (ii) the existence of very different residence times inside the reaction space due to the laminar flow regime. At the inner wall, both causes (maximum incident radiation and maximum residence time) add up to produce maximum conversion. Between both walls the incident radiation takes on an intermediate value (after some attenuation) and the residence time is minimum, producing the smaller conversion. Finally, close to the external wall, the radiation field reaches its minimum value (maximum attenuation) but the residence time is long and the resulting effect is an intermediate progress of the reaction. Reaction products show the opposite trends but, for this set of parameters, there exists a noticeable exception: at 5 = 0.5 (see Figure 16b),methyl chloride has already reached its maximum concentration and begins to act as one more reactant

2.4 I

_+--

Ind. Eng. Chem. Res., Vol. 34, No. 7,1995 2177

---1,-

0.2

I

2 0

0.1

-.'I-

I

1

I .4

12

0

I

I

I

JO

I.6

\ 1

0

I

I

I

I

I

02

04

0.6

08

1

0.8

I

5

Y

(a 1

(a) 03 f R0-

c

rRi = 2 cm

2

0.2

2

E

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0.1

n -0

0

02

06

04

08

I -

5

(b)

0

0.2

0.4

0.6

5 (b)

Figure 16. (a) Dimensionless radial concentration profiles for reactants and products at the reactor middle high position. (b) Dimensionless axial cup-mixing concentration profiles for reactants and products. [Reproduced from Cassano et al. (1994a) with permission. Copyright 1994 Research Trends.]

Figure 17. Dimensionless reactor cup-mixing temperatures as a function of dimensionless axial position. (a) Annular space equal to 1 cm. (b) Annular space equal to 2 cm. [Reproduced from

in a series reaction; Le., methyl chloride is consumed to produce di-, tri-, and tetrachlorinated products. In all the radial profiles the condition of impermeable walls (no heterogeneous terminating steps, as discussed previously) is observed, and the first derivative of concentrations is zero. Figure 16b shows the dimensionless cup-mixing concentration of reactants and products as a function of the dimensionless axial coordinate, under the same parametric conditions discussed in Figure 16a. Once more, different scales have been used for reactants (left side and plot with broken lines) and for products (right side and plot with solid lines). Both reactant concentrations decrease in a monotonous manner as a result of the reaction progress along the reactor length. Since the methane concentration has been used to define the dimensionless concentrations, its initial value is equal to 1,while the one corresponding to chlorine is equal to the feed molar ratio (F = 3). For these operating conditions, methane conversion at the reactor exit is close to 56%. Methyl chloride concentration increases in the first part of the reactor, up to approximately 5 = 0.4, where it reaches a maximum value and afienvard decreases, to give rise to a constant increase in the concentrations of all the other reaction products (methylene chloride, chloroform, and carbon tetrachloride). Surely, using a longer reactor or a different value for F , a similar maximum could have been observed for the di- and trichlorinated reaction products. These are two important operating variables for optimizing a prescribed selectivity.

In Figure 17 dimensionless reactor cup-mixing temperatures as a function of the dimensionless axial position are illustrated. Values are plotted for two different annular spaces: (Q, - Q) equal to 1and 2; the feed molar ratio is used as a parameter. Figure 17a presents the first set of values for F varying from 1to 9. For low values of the feed molar ratio (F = 1)the results show that the temperature decreases along the reactor length. These negative values of (z(f)) indicate that, under mild reaction conditions, the adopted constant refrigerating temperature (model assumption) is too low and the reacting mixture is cooled below the inlet conditions;ie., the reactor is overcooled. When F increases (F = 3) a temperature maximum is observed at the reactor entrance. When F is further increased, this maximum is more significant and moves toward the reactor middle positions (g x 0.3). For intermediate and high values of F, (df)) > 0 always, meaning that the temperature rises above inlet conditions. These maxima indicate the places where the refrigeratingsystem takes some form of control over the heating produced by the polychlorination reaction. For the operating conditions described in Table 8, it can be seen that at F = 9, the maximum increase in temperature is 32 K. For an annular space equal to 2 cm, Figure 17b presents an equivalent set of data but, in this case, temperature increments are very important. Temperature maxima are located closer t o the reactor exit and, for F = 9, an overheating as high as 156 K is obtained. This is an undesirable situation. In practice, evidently, it would be necessary to increase the linear velocity inside the reactor in order to bring conversions down t o limits

Cassano et al. (1994a) with permission. Copyright 1994 Research Trends.]


9

0

3

6

9

F Figure 18. Methane conversion and product selectivities vs feed molar ratio. [Reproduced from Cassano et al. (1994a) with permission. Copyright 1994 Research Trends.]

Figure 19. Three-dimensional representation of model predictions. Methane conversion as a function of feed molar ratio and annular space. [Reproduced from Cassano et al. (1994a) with permission. Copyright 1994 Research Trends.]

where the temperature can be better controlled. Obviously, this change will move the reactor operation outside the laminar flow conditions. Notably, this type of change will also contribute to enhance the heat transfer aptitudes of the gas phase by introducing turbulence and radial mixing (provoking a switch to much higher effective thermal conductivities). One can reach the conclusion that, in order to avoid undesirable side reactions, polychlorinations must be performed in turbulent flow regime and with low conversions per path. This conclusion is even more true if the annular gap is increased beyond 2 cm. In this case, the model will have to be changed to account for different flow and mass and energy transport conditions. In Figure 18 the model predictions are illustrated using the classical representation of conversions and selectivities vs the feed molar ratio. The annular space is 2 cm. The following results can be observed: (i)when F increases, the methane conversion increases; for our operating conditions, and F = 9, the methane conversion is almost 100%;(ii) methyl chloride selectivity is high only for very low values of F; an increase in F produces a monotonous decrease in methyl chloride selectivity; (iii)selectivity values for methylene chloride and chloroform show a maximum that depends on the value of F; for example, for methylene chloride the maximum selectivity is reached when F RZ 2 and, for chloroforin, maximum selectivity is obtained when F RZ 5; (iv)carbon tetrachloride selectivity increases monotonously when the value of F is augmented and, for example, values of F > 9 render a 100% selectivity for this species. Finally, Figure 19 portraits a three-dimensional representation of the model theoretical predictions. In the three-dimensional space defined by the axes (i) methane conversion [XCH,(%)I, (ii) feed molar ratio (F), and (iii) length of the annular space (Q - Q), one obtains a surface of response. This figure provides much information about the reactor performance. It gives the reactant conversion for an ample range of feed molar ratios (between 1 and 9) and different practical values of the annular space (between 0.5 and 3.5 cm). Again, the mean residence time is kept constant. Incidentally, one can mention that the methane conversion plotted in Figure 18 [(Q, - Q) = 2 cml is just one of the many curves that form the surface of response in Figure 19. Let us look at the results for a constant value of the annular gap. They are very similar to those observed

in Figure 18: increasing F, methane conversion increases. However, this increase is more pronounced when the annular space is larger (for the same mean residence time). Thus, for an annular gap equal to 0.5 cm and F = 9 the methane conversion is 70% while for an annular space equal t o 3.5 cm and the same F, methane conversion is close to 100%. This important change is only due to thermal effects on the reaction rates produced by the poor heat transfer aptitudes of the laminar flow operation; Le., when the annular space is increased, the reaction heat dissipation by the refrigerating fluid becomes less efficient. In other words, if the reactor could have been operated under isothermal conditions, an increase in the size of the annular space would have always produced a monotonous decrease in the product output concentrations. One can also make the analysis considering constant values of F and changing the dimension of the outer radius (always under non-isothermal conditions). For low values of F, it can be observed that increasing the annular gap produces just a mild increase in methane conversion, but these changes are much more significant when F is large. When F is large enough, an important increase in the outer radius renders the maximum methane conversion. Actually, beyond this point the output conversion diminishes; however, in our case this diminution is not very noticeable. In any event, in a region having high values of the optical thickness [both F and (IR,- Q) must be large1 methane conversion shows a sort of plateau where the reactor operates at very high conversions. One should expect that increasing the values of F and the annular gap beyond 9 and 3.5 cm, respectively, the output exit conversion will fall more dramatically. Physically, this behavior can be understood as follows: when the optical thickness is very high, the first part of the reactor (closer to the lamp) absorbs most of the incoming radiation; under these conditions, after a given distance along the radial direction the reactor becomes almost opaque to radiation entrance; in practice, portions of the fluid circulating beyond this radial distance do not receive radiation and pass through the reactor with reduced chances ,of undergoing reaction (recall that radial mixing in the laminar flow regime is not very important). The resulting effect is a decrease in the cup-mixing exit conversion. As indicated before, sometimes the heat transfer characteristics of

Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 2179 the reactor operation may disguise this effect due t o temperature rise effects on the kinetics of the involved reactions. The methane chlorination in the laminar flow regime is a good example where the reaction heat evolution prevails over the optical thickness effect for a rather wide range of operating conditions. With the same procedure, one could obtain response surfaces in terms of selectivities for each one of the products. These plots together with Figure 19 provide very useful information when, given a reactor, some product is of special interest; with these figures, one can obtain the best operating conditions to fulfill the desired requirements (the highest selectivity and the maximum yield). On the other hand, if the reactor must be designed according to some product specifications, one must use this type of simulations to decide about the lamp specifications, the reactor dimensions, and the appropriate initial concentrations.

useful wavelength interval [Imin,ImJ as that given by I I IT2 = For the most general case I m i n is defined as the lower practical wavelength bound (a wavelength where the value of the property is different from zero) of the reactant absorption coefficient, reaction primary quantum yield, lamp emission power, reactor wall transmission, and reflector reflectivity, whichever the largest wavelength. Similarly I m s is defined as the upper practical wavelength bound of the same properties, whichever the shortest. Let us illustrate the treatment of polychromatic radiation with the same example used before. Thus, the chlorination of methane but with a polychromatic radiation source will be considered. The emission of the lamp is discrete with several lines of emission. From eqs 22 and 23,

Ifin =Irl 5

XVI. Polychromatic Radiation Sources Up to this point we have been assuming that the reactor is operated with monochromatic radiation. Monochromatic radiation may be obtained with minor increases in the investment costs only for very particular applications, for example, when the so-called “germicidal” lamps are used. Under almost any other condition, getting monochromatic radiation implies a very significant increase in the cost of the used photons. Hence only very special product or process specifications may make advisable the adoption of monochromatic light. Conversely, for most usual applications polychromatic radiation will be normally used. In this section we will extend the design equations to make them suitable for the general case. It is convenient to introduce a small change in our treatment of the problem. Let us think in terms of wavelengths instead of frequencies; they provide a more familiar environment for the analysis. We will begin by defining the idea of useful wavelength interval. Let us consider a photochemical reactor operating under the following conditions: (i) the reactant absorbs radiation in a continuous way from I = I r 1 t o I = Ir2 having a molar absorptivity equal to aA(I); (ii) the reaction has a primary quantum yield equal to @& +p.,) that is k n o w n in the same wavelength interval (I = I r l to I = I r 2 ) ; (iii)the reactor wall is made of some material that has, for the adopted wall thickness, a transmission coefficient equal to (YR(I))which is different from zero between A = IRIand I = I R 2 ; (iv) a reflector is used which has an average (independent of the angle of incidence) reflection coefficient equal to (rRftk)) which is different from zero between I = and I = ARB; and (v) the lamp has a discontinuous emission with several lines of emission a t I = Ii ( i = 1,2,3, ..., n ) having an emission power equal to PA,^ ( i = 1, 2, 3, ..., n ) respectively; the wavelength range of the lamp ~ I =I L ~ . emission extends between I = I L and For most practical conditions a good design should (a) use a lamp that emits as much as possible within the range of the reactant radiation absorption, meaning that one must choose I L I I I r 1 and I L L~ Ir2; (b) use a material for the reactor wall that transmits the maximum possible of the emitted radiation, implying that (YdI)) must be as high as possible and at the same time I RI ~ Irl and I R 2 L I r 2 ; and (c) for the same reasons, use a reflector material having (rRkn)) as high as possible and ARfl 5 Arl and A R L ~ Arz. Under these conditions one can easily define the

In eq 113 we have used the Lambert-Beer approximation for the absorption coefficient and

For simplicity we have written @prim,n = @I, and (YR(I)) = Yn. In this system there is no reflector ((rRftn)) is not used) and inerts and products do not absorb radiation. Let us define the following characteristic variables:

P, = Jam= P*dA Amin

(115) (116)

(117) Yc=-J1

p,

Amax &nin

YAPAdA

(118)

and for completeness, (119)

Also,

then:


The summation must be performed over all the spectral lines of the lamp emission spectrum for which aA, @A, and Yi are different from zero. Then

In eq 122 we have

which is a dimensionless quantity and may be thought as the ratio of a mean residence time and a characteristic initiation reaction time; we call it the initiation number. A second important dimensionless number has been defined in eq 120 which is the absorption number A. It provides an idea of the optical characteristics of the reacting system; it is easy to recognize in that it includes all the variables involved in a conventional absorbance, Le., an absorptivity, a concentration, and a length. Thus, if A is high, we will be dealing with a reactor that is optically thick, and if it is low, the reactor will be optically thin. Also, we have used

(124)

Pi = PJP,

(128)

Let us concentrate on the wavelength dependent integral:

In our case, absorption by chlorine as well as the reaction quantum yield and the reactor wall transmission are continuous functions of wavelength. On the other hand, the voluminal emission, arc lamp provides an emission spectrum that is discontinuous; let us suppose that it is a mercury, medium-pressure, arc lamp. Let us write

Pi = P,d(A - 1,)

(130)

with 6 being the Dirac function. A finite number of monochromatic lamps, each emitting the output power corresponding to each line of emission (P,), can now substitute for the polychromatic lamp. For this particular case, the interval where the lamp has emission lines is wider than the one corresponding to the absorption by chlorine. Also the wall transmission (for example, if the reactor is made of quartz) is very high for an even wider wavelength interval and @A is known in the whole interval of radiation absorption. Hence 1 m i n and 1" are defined by the chlorine absorption range. Then, eq 130 can be written as rp=n

Here, P,, q,a,, and Y, are the lamp output power, the chorine absorptivity, the reaction primary quantum yield, and the reactor wall transmission at 1 = A,, respectively, and n is the number of emission lines produced by the lamp between 1 = Amin and 1= 1 m u . The final equation for the rate of initiation using polychromatic light results

This derivation was made for a lamp with a discontinuous emission, i.e., a lamp that produces lines of emission at given wavelength positions (Q, = 1 to p = n). There are lamps that produce a continuous emission. In this case the interval of continuous emission (from 1 m i n t o Am=) may be turned discrete dividing the said interval in a number ( n )of small intervals of the emission spectrum. In the previous derivation we can substitute each line of emission Q, by an emission interval Aq, where A q corresponds to each of those wavelength intervals used t o render discrete the continuous emission ( A q takes on values from 1 to n). Thus, when the emission is continuous, n represents the number of intervals used for turning discrete the continuous curve of emission of the lamp. For each interval AQ,,the Q, index refers to the mean value of each wavelength interval and provides the specific wavelength where all the wavelength dependent reaction and a, Y,, P,, reactor properties must be computed (q, and r,). In general, continuous emission is produced by lamps having a superficial emission; a typical example is a fluorescent lamp. Applications of the above systematic treatment for photochemical systems using polychromatic light can be found in Claria et al. (1988), Alfano and Cassano (1988a,b), and Cabrera et al. (1991a-c). In particular, Cabrera et al. (1991~)used a lamp with superficial emission. M I . Modeling of Reacting Systems Employing Reflectors In some cases, photochemical reactors are irradiated with the aid of a reflector. Figures 20-23 illustrate typical cases. In the first case the reflector is a cylinder of elliptical cross section. A tubular lamp is placed parallel to the elliptical cylinder a t one focal axis and the tubular reactor is located a t the other axis. The characteristics of the elliptical surface make rays emerging from the lamp concentrate on the space occupied by the reactor. Direct irradiation of the reactor by the lamp is usually not too significant. This type of reactor was initially proposed by Baginski (1951) and has been widely used ever since, particularly in bench scale studies. Significant contributions to the modeling of this system can


cs_T"I

,LAMP

LAMP

REACTOR

{ELIPTICAL REFLECTOR

Figure 20. Continuous flow, tubular reactor inside a cylindrical reflector of elliptical cross section. [Adapted from CerdA et al. (1973) and ClariA et al. (19881.1

Figure 23. Single lamp-multitube, continuous flow photoreactor. [Adapted from Tymoschuk et al. (1993a)J

Figure 21. Perfectly mixed, semibatch, cylindrical reactor irradiated from the bottom by a tubular source and a parabolic refledor. [Adapted from Alfano et al. (1985) and Alfano and Cassano (1988a)J

Figure 22. Single tube-multilamp, continuous flow photoreactor. [Adapted from Alfano et al. (1990).1

be found in the work of CerdA et al. (1973, 1977) and De Bernardez and Cassano (1982). In the second case, the reactor (actually of any desired shape) is irradiated from the bottom by a tubular lamp and a parabolic reflector. A similar system can be used, for example, for irradiating a flat plate reactor from one

side. The tubular lamp is located at the focal axis of the cylindrical reflector that has a parabolic cross section. Once more both, the lamp and the reflector, have cylindrical shapes and the axis of the tubular source is parallel to the generatrix line of the reflector. In this case, contributions from direct irradiation are significant. The characteristics of the parabolic surface help concentrate the radiation coming out of the lamp in the reaction space. The modeling of this system was developed by Alfano et al. (1985, 1986c,d). An extension of the concepts used to describe the performance of the elliptical and the parabolic reflectors was employed in modeling a multilamp-single tube reactor. As shown in Figure 22, a cylindrical, tubular reactor is surrounded by several lamps (in the figure only four). Each one of the lamps has one reflector that can have different cross sections. Alfano et al. (1990), Vicente et al. (1990), and Esplugas et al. (1990) investigated the radiation field produced in these type of reactors. Finally, Figure 23 shows a multitube-single lamp reactor. In this case, a tubular lamp is surrounded by several tubular reactors. The reactor tubes, in their turn, are surrounded by a reflector of circular cross section. This reactor was conceived for operating photochemical reactions under pressure. As indicated in the figure, several tubes with a diameter reduced to the point of withstanding the desired pressure, made of quartz or glass, could be arranged to form a pseudoannular reacting space surrounding the tubular lamp. The system was investigated by Tymoschuk et al. (1993a,b). We have seen eqs 96-99 that the radiation source power (PV,&and the lamp characteristics (LLand RL) enter directly into the modeling of the reactor a t the moment when the reaction kinetics is written and an expression for the rate of initiation is needed; i.e., the key, distinct characteristic of this type of processes lies almost exclusively in the activation step produced by radiation and this phenomenon only affects the formulation of the initiation kinetics. An identical consideration applies to the effect of reflectors. The existence of reflecting devices puts some limitations on the shape and configuration of the reacting system, but leaving aside this aspect, the characteristics of the reflector


enter into the reactor modeling process exclusively in the formulation of the rate of initiation. An imaginary observer placed a t a point inside an annular type of reactor (with no reflector, Figure 1) would “see”the lamp; in the most general case a portion of it, although from some parts of the reactor he could see the whole lamp. This concept of “the part of the radiation source seen by an observer from an arbitrary point inside the reactor” has been used t o complete the derivation of the emission model for the direct irradiation produced by a tubular source. In fact, it can be noticed in the detailed derivation of the emission model that the boundary condition for the radiative transfer equation applied t o the reactor is obtained by application of these ideas; the same rationality lies behind the procedure used to obtain the limiting angles of integration for computing the contributions of the lamp emission to each point in the reacting space (solid angle of reception). The existence of a reflector (a mirror, with specular reflection) poses a very similar, although much more complicated problem to evaluate its effects on the reacting system. In short, one needs to use the principles of geometric optics to know the shape and size of all the images of the lamp that can be seen, on the reflector surface, by an observer placed at any arbitrary point inside the reactor. In other words, ifa point inside the reactor receives radiation from the reflector (reflected radiation), it is because an imaginary observer located at such a point would see one or more virtual images of the lamp on different parts of the reflector surface (the mirror). Then, working with the idea of lamp images, instead of the lamp itself, one can develop the set of equations that are required for obtaining the boundary condition and the limiting angles for integrating the radiative transfer equation inside the reactor, but, in this case, for contributions due to reflected or indirect radiation. Obviously, then, at each point, the total incident radiation will be the sum of direct and indirect radiation. Let us look a t the problem for homogeneous systems; the extension to pseudo-homogeneousreactors is almost trivial. A slightly different form of eq 57 will be the starting point. Actually, a radiation balance at the same point inside the reactor as the one used in deriving eq 57 would not recognize if the arriving radiation comes directly from the lamp or if the radiation bundle has been previously reflected. The only difference will be that in the process of specular reflection the radiation ray suffers a loss of energy, and this effect is accounted for by means of a reflection coefficient, rRf,v(Q), which is a number smaller than 1. The reflection coefficient is a function of frequency and the angle of incidence. For reflected radiation, eq 57 may be written as follows:

The boundary condition [t(6,4)1~f and the limits of must be obtained. integration 6Rd4) and 1. Boundary Condition. A lamp with voluminal emission will be considered. A similar reasoning can be applied for lamps with superficial emission. The procedure is exactly the same as that represented by eqs 71-88. The only important difference appears when i.e., eq one needs the values of ~2(x,6,4)and Ql(x,6,4); 86 no longer applies. Results will be obtained for any

m REACTION

F

Figure 24. Geometry and coordinate systems for reflected radiation. [Reproduced from Cassano et al. (1994a) with permission. Copyright 1994 Research Trends.]

form (cross section) of cylindrical reflector, for any form of reactor, and for tubular lamps. Let us start from a point of incidence I inside the reactor; as indicated in Figure 24, relative to a fured Cartesian coordinate system (XF,YF, ZF) its location is given by (XI, y ~ 21). , Notice that the ZF coordinate is always chosen parallel to the lamp and the reflector cylinder axes. An arbitrarily chosen ray arrives at point I with a direction (0, 4) from the three-dimensional space. Placed a t point I one may follow the trajectory of the ray in a reverse way: from the point of incidence I to the point of reflection (point P) and from the position P to the radiation source. This particular ray crosses the lamp at points El and E2. The point I is irradiated (by reflected radiation) in the direction (0, 4) by the emitters of the voluminal lamp that are located from E2 to El. With this idea in mind one may proceed as follows: 1. Identify the ray that arrives at point I with an arbitrary direction (e,#) described in a mobile, spherical coordinate system located at point I. A unit vector €1 representing this direction is given by the following expression: cI = sin 8 cos 4 i

+ sin 6 sin 4 j + cos 8 k

(135)

2. Identify the point P on the mirror. This is the point of reflection. The coordinates of this point ( x p , yp, zp) relative to a mobile, Cartesian coordinate system ( x , y, z ) located at I are xp = eI sin 0 cos 4 yp = QI sin 0 sin ql

zp = eI COS e (136) The position of point P will be precisely determined if one can obtain the value of = PI. It can be obtained from the following: 2.1. the equation of the straight line that goes through the point I and has direction ( 6 , 4) written in the ( x , y, z ) system 2.2. the equation of the reflecting surface written in the same system (relative to the point I). The point of intersection of both equations provides the expression for calculating QI. In the most general case it will have a form as follows:

eI = @,(e,4, reflector parameters, xI,yI, zI)

(137)


h = &($,

reflector parameters, xI,yI,2,)

(144)

In the figure, values having a prime represent projections of the variables on a (XF,YF) plane; they are very convenient for graphic representation and computational purposes. 5. Identify the points E 1 and E 2 corresponding t o the intersection of the ray with direction EE with the cylinder representing the lamp. These two points will be located at xE(1,2)

= @E(1,2) sin

YE(1,2) = @E(1,2) sin

cos $E sin h

REFLECTION

Figure 25. Projections of variables for reflected radiation on a ( x y )plane. [Reproduced from Cassano et al. (1994a) with permission. Copyright 1994 Research Trends.]

In any computation, the position of point I is k n o w n (a point inside the reactor where the reaction rate must be calculated), the reflector geometric parameters are known, and the direction of the ray must be chosen from all possible directions of incidence (and also they will be known). Actually, the limiting values of (8, 4) for choosing all possible directions of arrival at point I will be treated as a second part of this section. Consequently (XP,yp, zp) will be known. 3. Identify the unit vector normal to the reflector at point P. Let us call it n p . If the reflecting surface has a generic equation represented by

flxy)= 0

= npA

(140)

or

j

k

np,

npy

0

'E+

'Ey

=

i

j

k 0

np,

npy

€1,

€ 1 ~€1,~

(141)

Equation 141 is not enough to obtain the three components O f €E. But since it is a unit normal vector, we can use another equation:

The result representing this vector may be written as = sin 8 cos q5E i

(145)

However, @E,1 and @E,2 are not known. Their value can be obtained from the following: 5.1. the equation of the straight line that goes through the point P and has a direction represented by €E, written in a Cartesian coordinate system with origin at the point P and the z axis parallel to the lamp axis 5.2. the equation of the lamp cylinder written in the same system (relative t o the point PI;this produces a quadratic equation. The point of intersection of both equations provides the expressions for calculating @E,1 and @E$ As expected, two values of the distance are obtained. In the most general case they will have a form as follows: @E(1,2) = @E(1,2)(0, $9

reflector parameters,

YI,zI?

+ sin 8 sin (PE j + cos 8 k

rL)

(146)

6. From eqs 137 and 146, the values of e1 and e 2 for reflected radiation will be obtained as follows:

It must be noticed that in both eq 138 and eq 139 the z coordinate is not necessary. This is because our reflector is a cylinder and the tubular lamp is placed parallel to the generatrix line of such a cylinder. 4. Identify the components of the unit vector in the direction of the radiation ray before reflection at point P, €E. It can be obtained from the law of specular reflection:

i

= @E(1,2) cos

(138)

the unit normal vector t o this surface is given by

npA

'E(1,2)

(143)

In eq 143, the value of C$E (see Figure 25) will have the following generic form:

and finally,

which will result in an expression of the form:

rA@s]Rf = [AesIRf(8,$,

reflector parameters, xI, yI,zI,rL) (149)

Since the reflector parameters are k n o w n and the lamp radius is also known, for any point I inside the reactor and any chosen direction of radiation arrival the value required to obtain the boundary condition will of [AQJR~ be always known. The equivalent version of eq 81 takes now the following form:

Very often and for simplicity, an average value over all directions may be used for the transmission and reflection coefficients; but since computations are carried out for every direction, the directional dependence of the reflection coefficient can be incorporated into the evaluation of the boundary condition without great difficulties.

/->

2184 Ind. Eng. Chem. Res., Vol. 34,No. 7,1995 REACTION

\ SURFACE OF REFLECTION

Figure 26. Projections of variables for reflected radiation on a (y?) plane. [Reproduced from Cassano et al. (1994a) with permission. Copyright 1994 Research Trends.]

The only remaining question is which are the values that the angles (e,#) can take? The answer is provided by the limiting angles for reflected radiation that are needed to integrate eq 134. They will be obtained in what follows. 2. Limiting Angles for the Variable 8. According to Figure 26 the limiting angles for 8 can be obtained from

(151)

(152) These equations can be expressed in terms of the distances projected on the plane perpendicular to the ZF axis [plane (XF,~ ~ 1 1 :

e{(#) = eI sin 8

and

eE’(#) = @E sin 8

(153)

which renders

In eqs 154 and 155 ZIis measured from the lower lamp end. 3. Limiting Angles for the Variable 9. The ideas are the same as those used in deriving eq 95 but considering reflected rays. Then, the limiting angles correspond to those values of 4 that define tangent points t o the lamp boundary. These angles can be derived from the condition fulfilled by the tangent lines to the lamp surface; i.e., = @E,2(e7#)

(156)

- @E,2’(4) = 0

(157)

or which is the same @E,1’(#)

Equation 157 renders a nonlinear, algebraic equation implicit in the variable 4. It can be solved numerically and the limiting values of 41 and 4 2 so obtained.

From the operational point of view one proceeds in the following manner: (i) Calculate 41 and 4 2 with eq 157. (ii) Calculate el(@)and &(4)with eqs 154 and 155 for all the valid interval of # [according to (ill. (iii) Calculate [ A Q ~ ( O , ~ ) Iwith R ~ eq 149 for all the valid intervals of 8 and 4 [according to (i) and (ii)]. (iv) Calculate the boundary condition according to eqs 150 and 84 and the results from (iii). (v) For the given point I, solve the integral represented by eq 134 using the boundary condition obtained in (iv) and the limiting angles provided by (ii) and (i). This methodology has been applied with remarkable success to photochemical reactors using different reflector systems. To show the specific results for each particular application, one would have to resort to a detailed and lengthy treatment, well beyond the scope of this paper. In what follows, four different systems that have been modeled (and have received experimental validation) will be commented on briefly. A short description will provide the highlights for each of them. 1. The first system is a cylindrical reactor inside a cylindrical reflector of elliptical cross section. The system is represented in Figure 20. The reactor is irradiated with direct and indirect radiation. The main advantage of this system is the total isolation of the lamp from the reactor. Although it is difficult to conceive its use in large scale operations, it is very suitable for bench scale work. The elliptical reflector permits the irradiation of the reactor from outside, but one must be aware that only under very special conditions is this irradiation uniform. The main reasons for this nonuniformity are (i) the effect of direct radiation, (ii) the finite size of the lamp (TL t 0)) and (iii) the obstruction produced by the nontransparent reactor that eliminates part of the reflector surface from the possibility of reflecting radiation beams (with an opaque reactor, radiation that reaches the reactor in a direct way and is absorbed cannot be reflected). It has been shown that angular symmetry can be obtained when (i) the distance between foci is large, (ii) the ellipse eccentricity is small, and (iii)the radius of the reactor is small (De Bernardez and Cassano, 1982). Cerdti et al. (1977)modeled the system including the effects of umbra and penumbra produced by the opaque reactor as well as the effects of successive reflection processes after the first one. Under normal operating conditions, in bench scale sizes, the main conclusions are (i) the first reflection accounts for 85-95% of the total incident radiation on the reactor surface, (ii)direct radiation accounts for 5 4 % of the total radiation arriving t o the reactor, and (iii) generally, successive reflections after the first can be neglected. Very good results were obtained with this reactor in modeling studies performed by Claria et al. (1986)for the photodecomposition of oxalic acid solutions, Claria et al. (1988)for the chlorination of ethane, Cabrera et al. (1991a,b)for the chlorination of methane, and for the chlorination of methyl Cabrera et al. (1991~) chloride in the liquid phase and under pressure. In all cases, the kinetic constants were taken from data published in the physicochemical literature and no adjustable parameters were employed. 2. The second system is a tubular lamp inside a cylindrical reflector of parabolic cross section, irradiating the surface of radiation entrance to the reactor. A second type of reflector was used, for example, in studies related to gas-liquid systems (Figure 21). It

Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 2186 consists of a well-stirred tank reactor irradiated from the bottom. Once more the lamp system is independent of the reactor. Actually, it could be used to generate an irradiated region above the reflector regardless of the shape of the reactor; i.e., the cylindrical reactor of the figure can be substituted, for example, by a flat plate. The radiation field inside the reactor of the figure was modeled by Alfano et al. (1985, 1986c,d). It was found that (i) as expected, the radiation field decreases when the axial coordinate increases; (ii) a similar effect can be observed for changes along the radial coordinate; however, there are points located in the central zone of the plate where changes are almost negligible; (iii) as far as the angular coordinate is concerned, azimuthal asymmetries may be significant only for points very close to the reflecting system and they tend t o vanish when the axial coordinate increases; and (iv) direct contributions from the lamp account for 3 0 4 0 % of the total irradiation produced by the system (the rest of it corresponds t o reflected radiation). One can conclude that even though the tubular lamp with the parabolic reflector does not generate a completely uniform field as far as the angular and radial coordinates are concerned, there exist geometrical arrangements where only the unavoidable axial variations will be significant. This reactor seems to be very appropriate for reactions where vigorous stirring is important (for example in gas-liquid reactions) but it does not provide a practical solution for large scale operations. Conversely, the parabolic irradiating system used in combination with a continuous flow, flat plate reactor, of reduced thickness, could be a suitable design particularly for reacting fluids with significant absorption (an optically thick reacting medium). For a gas-liquid system this device was successfully used by Alfano and Cassano (1988a,b). They studied the consecutive photochlorination of trichloroethylene operated in a semibatch fashion, with very good agreement between experiments and model predictions. 3. The third reflector system is a multilamp-single tube reactor (Figure 22). Alfano et al. (19901, Vicente et al. (1990), and Esplugas et al. (1990) investigated the radiation field produced by reflectors of cylindrical, elliptical, and parabolic cross sections. The main conclusions were (i)with four lamps, angular asymmetries in the reactor cross section are almost negligible and consequently, the reactor can be modeled with just two-dimensional mass and thermal energy balances; (ii) reflected radiation provides between 60 and 70% of the total irradiation and consequently the reflector design has a significant effect on the total yield, and (iii) reflectors of elliptical cross sections produce the best yield. This is a very practical reactor for continuous operation. Assembling several of these reactors in series, one can provide the required reaction time for any desired flow rate. Devices of this type having from three to eight tubular lamps have been used for water purification purposes. 4. The fourth reflector system is a multitube-single lamp reactor (Figure 23). The system was investigated by Tymoschuk et al. (1993a,b) and the main conclusions are (i) reflected radiation does not provide an important contribution; (ii) inside the reactor, angular asymmetries are always very important and hence, three-dimensional mass and energy balance equations should be used for the reactor

model, and (iii) in a well-designed system (with the optimal number of reactor tubes and optical thicknesses) the effects of reflected radiation can be normally neglected. This system provides a very practical arrangement for any size of application. It permits (i) simple maintenance procedures, (ii) facility for cooling or heating the reactor tubes, (iii) a variable throughput, and, if desired, (iv) operation under pressure.

XMII. Applications to Homogeneous Systems Up to now we have presented the known rigorous theory on photoreactor analysis and design. Along its development it has been indicated that several experimental validations have been conducted. Some of them will be briefly described in what follows. 1. Annular Photoreactor with Simple Kinetics. This first application presents an a priori modeling of a well-known reaction carried out in a continuous annular photoreactor (Figure 1). The radiation field was described by resorting to the extended source model with voluminal, isotropic emission (E-VIE model) proposed by Irazoqui et al. (1973). On the basis of the kinetic data published by Leighton and Forbes (19301, Forbes and Heidt (1934), and Heidt et al. (1970), the photodecomposition of the complex formed by oxalic acid and uranyl sulfate in aqueous solution was modeled. Experiments were carried out under isothermal conditions. The overall chemical reaction has a product distribution that is dependent on the pH and it may be written as follows

(UOT)* + H,C204

+

= +

UOzZ+

(UOzZ+)* H2C204 UO;+

+ CO + CO, + H20 + HCOOH + C02

The reactor was modeled with the momentum, mass, and radiation energy conservation equations described in sections 111and XI. The kinetics was formulated with a phenomenological approach, following the oxalic acid concentration and using an overall quantum yield according to

The reaction rate was calculated on the basis of the published quantum yields. The optical properties (absorption coefficient of the uranyl oxalate complex) are almost independent of the reaction extent as long as the conversion is kept below 20%. However, since we are not dealing with a perfectly mixed system but a laminar flow reactor, the hypothesis of using a constant absorption coefficient loses validity. This is so because it is inevitable t o reach local values of conversions that are well above 20% (at the reactor wall closer to the radiation source). At wavelengths below the n e a r - W range (preciselywhere absorption is stronger), when the oxalic acid-uranyl sulfate molar ratio falls below 4, the absorption coefficient becomes a nonlinear function of the oxalic acid concentration. In practice, when the local conversion is higher than 20%, the kinetic behavior shows a dependence upon the concentration of the oxalic acid, because the nature of the uranyl oxalate complex changes and so does the absorption coefficient (De


0’50

m

X

Xpred

Figure 27. Experimental conversion vs predicted conversion for uranyl oxalate decomposition. Open symbols, reactor no. 1 (di,= 4.6 cm, do, = 5.4cm, LR = 30 cm); filled symbols, reactor no. 2 (din= 4.6 cm, do, = 7.2 cm, LR = 30 cm): (0, U) computed with lamp maximum output power, (A,A) computed with lamp average life-span output power. [Adapted from Cassano and Alfano (19911.1

Bernardez and Cassano, 1985). Thus, in this case, the coupling “radiation absorption-reaction extent” (section XIII) exists. The rigorous approach also requires the computation of reactor and lamp wedges (the irradiated space is longer than the lamp length). The three-dimensional characteristic of the emission process causes the reactor to be partially irradiated in the incoming and outgoing regions, below and above the actual lamp length (Romero et al., 1983; De Bernardez and Cassano, 1985). This fact increases the effective reaction volume. This effect is more noticeable when the inner reactor wall is close to the lamp. The expressions for the &limiting angles should be modified to account for the existence of these partially irradiated regions; the reader is referred to Romero et al. (1983)for a detailed analysis of the reactor wedge computations. Similarly, calculations must be corrected to account for the effective lamp length (effect of the lamp electrodes on the upper and lower limits of integration). If no corrections were introduced we would be computing as lamp emission volumes portions of the cylinder that correspond to regions in the space where there is no emission at all. This effect is also significant when the inner reactor wall is very close to the lamp; for a exhaustive description, the reader may refer to De Bernardez and Cassano (1985). We performed the experimental verificationswith two reactors, the characteristics of which are indicated in Figure 27. We employed a 15W input power germicidal lamp and an aqueous solution of 0.005 M oxalic acid and 0.001 M uranyl sulfate. Neither free nor experimentally adjustable parameters were used. Details on the modeling, the apparatus, the operation, and the obtained results may be found in De Bernardez and Cassano (1985). Figure 27 illustrates the quality of the predictions compared with the experimental results for both reactors. In the figure, the open symbols correspond t o reactor no. 1 and the filled ones to reactor no. 2. Squares correspond to predictions computed with the lamp maximum output power supplied by the manufacturer, and triangles are the values using the average life-span output power value, also provided by the lamp manufacturer. It must be stressed that, a t low wavelengths, the observed good agreement can be obtained

only when a variable absorption coefficient (a function of the local values of the oxalic acid concentration) was used. 2. Cylindrical Reactor inside an Elliptical Reflector. 2.1. Monochlorinationof Ethane under Isothermal Conditions with Polychromatic Radiation. This section presents the results of an a priori modeling of a complex chain reaction, carried out in a system where direct and reflected radiation are present (Figure 20). We performed the monochlorination of ethane in the gas phase, using an excess of nitrogen as an inert. The reactor was a tubular, continuous flow system operated under isothermal conditions. The kinetic mechanism and the values of the specific rate constants were taken from the literature. They comprise values for homogeneous and heterogeneous (at the wall) reactions (Ivin and Steacie, 1951; Noyes and Fowler, 1951; Ivin et al., 1952; Steacie, 1954; F’ritchard et al., 1955; Gosselain et al., 1956; Knox and Nelson, 1959; Kerr and Trotman-Dickenson, 1960;Ayscough et al., 1962; Chiltz et al., 1963; Hutton and Wright, 1965; Clyne and Stedman, 1968; Clark and Clyne, 1970; Davis et al., 1970). The full reaction sequence is shown below: initiation c1,

+ hv

propagation

+ EtH Et’ + C1,

Cl’

-

-

-

2cl’

0.01 m heterogeneous termination reactions do not have significant effects on the overall rates. Ratios of reactor volume to reactor “wetted” area larger than 0.01 m are easily achieved in practice. Claria et al. (1988) also showed that in these systems the local or microscopic steady-state approximation for intermediate unstable species can be applied without error. In general it is required that the time needed to reach an “almost steady-state” value for the free radical concentrations be a small fraction of the mean residence time spent by the reactants inside the reactor. This condition depends upon many variables; the most important ones are grouped in the dimensionless number J = [@c~YJ‘,V [LL(v,)](eq 123). When J is large, the mssa applies. Practical values of J may range from 10-1 to lo-*. We have found that for J > lo-, the mssa applies without difficulties. For most practical situations, if the mean

Ind. Eng. Chem. Res., Vol. 34,No. 7, 1995 2187 residence time is longer than 10-1s, the hypothesis can be safely used. Large values of ea favor its application. Very fast overall reaction rates, on the contrary, may require some precautions in its use (because for a given conversion the system would require very short mean residence times). The reaction kinetics was described following a procedure equivalent to the one used in section XIV. De Bernardez and Cassano (1982)developed design criteria under which angular asymmetries in the cylindrical reactor can be neglected. The key parameter was The established conditions [small ellipse the ratio dr~. eccentricity (e I0.41, large value of the distance between , rp,/rL < 0.51 can only be achieved in foci (c > 4 0 r ~ )and bench scale equipment. For large scale reactors the assumption is only an approximation. Angular symmetry was assumed here because the above requirements were met. Under these conditions the mass balances can be simplified because they may be mitten in two dimensions only. The reactor was modeled under the same set of assumptions listed in section 111, but eqs 1 and 4 must be replaced by

The dimensionless radius is now defined according to

y = rlrR

(161)

The mass balances are identical to eq 5 but the second boundary condition (eq 8) must be changed. For stable species,

Under isothermal conditions, the thermal energy balance is not required. However, the radiation field is now considerably more complicated than in the case of the annular reactor. Radiation arrives at a point inside the reactor by two different mechanisms: (1) direct radiation from the lamp to the reactor; (2)indirect radiation after reflection on the cylindrical reflector. It should also be noted that, strictly speaking, indirect radiation may be the result of one or more than one reflections of the same ray. Cassano et al. (1986)have shown that when curved reflecting surfaces are present extended (three-dimensional) emission models must be used. In this case, a nonfluorescent, arc lamp was employed and the E-VIE model was applied. Direct radiation was modeled as indicated for the annular reactor because the physical situation is identical; i.e., the presence of the reflector does not affect the transport of direct radiation from the lamp t o the reactor. Results were obtained with an almost straightforward application of the equations presented in sections VII, IX,and X. Reflected radiation was modeled as described in section M I ; the resulting equations applied to the cylindrical reflector of elliptical cross section can be found in Cerdd et al. (1973, 1977) and Clarid et al. (1986, 1988). The following additional hypotheses were used (i) The reflector is a perfect elliptical cylinder.

(ii) The lamp is located in such a way that its centerline passes through one of the focal axes of the elliptical reflector. (iii)Specular reflection occurs with an average reflection coefficient that is independent of wavelength and direction. (These restrictions could be easily relaxed and were used here for simplicity.) (iv) The reflected radiation comes only from the elliptical reflector (Le., the top and bottom parts of the cylinder do not reflect radiation). (v) Transmittance of the reactor wall is represented by an average coefficient that is only a function of frequency. This example can be used to illustrate the selection of the useful wavelength range when polychromatic radiation is employed (section XVI). For this system we had the following situation: the lamp emission spectral distribution was discrete and lines of emission are significant between ALJ = 200 nm and A L , ~= 1400 nm. The transmission coefficient YR,”for quartz has a value that may be considered constant between L R , ~= 220 nm and &,2 = 2000 nm (below and above these two limits, values are no longer independent of wavelength). The average reflection coefficient for specularly finished, aluminum reflectors is important (but not constant) from h , l = 160 nm to h , 2 > 800 nm. Absorption of radiation by chlorine is important from = 250 nm to Ar,2 = 500 nm. These limits indicated that one can restrict the range of analysis from = 250 nm to Amax = 500 nm. Within this range, the treatment of polychromatic radiation was done according with the procedure described in section XVI. Since the operation was almost isothermal (a large excess of nitrogen was used and conversions never exceeded 60%), the rigorous solution of the problem involves the numerical integration of a set of five integro-partial differential equations corresponding to the stable species mass balances. The design parameters were taken from the literature (kinetic constants, reflection coefficient for aluminum, molar absorptivities for chlorine, physical properties of reactants and products, etc.), or from the manufacturer’s specifications (lamp characteristics, reactor and reflector dimensions, for example). The model was verified in bench scale experiments with an emitting system made up of a lamp with 30 W of nominal input power and a reflector built with an aluminum reflecting sheet with Alzac treatment. The reactants were carefully purified and the reactor operation was minutely controlled. Each experimental datum demanded a minimum of 8 h of operation. More details about the modeling, the experiments, and the results may be found in Claria et al. (1988). Figure 28 portrays the a priori predictions compared with the experimental results. The agreement is very good. 2.2. Non-Isothermal Polychlorination of Methane with Polychromatic Radiation. In this case we will show the experimental verification of a model representing the conjunction of a difficult irradiating system as the elliptical reflector and a complex kinetics represented by the successive photochlorination of methane, where attractive selectivity studies can be performed. The tubular, circular cross section, continuous flow reactor performance was not isothermal. The system was explained in detail for the example described in section I11 but for an annular reactor. In addition t o the changes already mentioned in eqs 159-


/l

1

0.80

A

0.45

x-

c I

t I

0.30 I/ 0.30

d

/. I

X"

/

i I

I

0.45

/I 0.60 pred

X pred Figure 28. Experimental conversion vs predicted conversion for monochlorination of ethane in the gas phase. d~ = 0.4 cm, LR = 30 cm, % Clz = 1-7. [Adapted from Cassano and Alfano (1991).]

Figure 29. Experimental conversion vs predicted conversion for polychlorination of methane. LL189A-10/1200 W lamp, (0)360 UA-3 lamp, (A)TLK 40/09N lamp. [Adapted from Cassano and Alfano (1991).]

1

162,eqs 12 and 13 must be replaced by 325

In this case, with the simplifications indicated in section 111, the number of simultaneous integra-partial differential equations to be solved has been increased to seven. Three different lamps have been used, all of them with very different output powers and emissions having a very wide range of polychromaticity; additionally, one of the lamps is of the fluorescent type (with superficial emission) and it required the using of the E-SDEsource model. The lamps used in the experiments had the following input powers: 40, 360, and 1200 W. The runs were performed on bench scale equipment a t three different nominal temperatures (297.9,312.8,and 322.2 K). The feed molar ratio of chlorine to methane was changed from 0.5 to 10. Thus, the experimental verifications were performed under a broad range of operating conditions. Not only total reactant conversions but individual stable product concentrations (selectivities) were subjected to testing. Figure 29 shows a compendium of all the experimental data, compared with the model predictions. The model also provides the complete temperature field. Figure 30 shows, for example, the mixingcup (radial average) axial temperature profiles for runs performed with the three lamps when the feed molar ratio was 9.5. Further details on the subject can be found in Cabrera et al. (1991a,b). The model was also used t o study the non-isothermal photochlorination of methyl chloride in the liquid phase and under pressure (450kP)with polychromatic radiation. Carbon tetrachloride was used as a solvent. Reaction constants were taken from the known reaction mechanistic kinetics in the gas phase. Both, reactant conversion and product selectivity, were experimentally verified and the agreement was good. Significant temperature and concentration gradients inside the reactor were found. The details can be found in Cabrera et al. (199IC). 3. Multilamp-Single Tube Photoreactor with Cylindrical Reflectors. Simple Reaction. This example shows the modeling of a commercially available

l

_______________--------

__-e-

295 _. .

0.0

1

I

I

I .o

0.5

5 Figure 30. Reactor cup-mixing temperatures as a function of dimensionless axial position. (-) P, = 1200 W, Pc = 360 W, (- - -1 P, = 40 W. [Adapted from Cassano and Alfano (1991).1 (*a*)

type of reactor used for water and wastewater treatment. The system has a cylindrical quartz tube irradiated from outside by cylindrical tubular lamps symmetrically located about the reactor. One cylindrical reflector of circular shape surrounds each lamp (normally, LL = LRf). Typically, as in this example, four germicidal lamps are employed (Figure 22). Thus, each point inside the reactor receives radiation from four lamps and four reflectors. Usually the reactor is considerably longer than the lamp and the reflector, giving rise to the existence of important regions in the entrance ( z o , ~and ) exit ( ~ 0 , portions ~) of the tube that are partially illuminated due to the spherical modalities of the lamp emission (0 Iz (for reaction) IZ0,i LL

+ +

ZO,e).

According to the physics of the problem direct and indirect contributions from the four radiation sources must be added. However, at each plane (r, ,f?) of the reactor, it is not required to compute the whole radiation field. Clearly, symmetry considerations allow us to reduce the analysis to a region limited by 0 Ir -= m,0 5 /3 < ~14, and 0 Iz 5 Z0,i LL ZQ. Additionally, an approach was devised which reduces the solution of the problem composed by one reactor, four lamps, and four

+ +

Ind. Eng. Chem. Res., Vol. 34, No. 7,1995 2189 reflectors to the analysis of an equivalent system formed by one reactor, one lamp, and one reflector only. With this procedure the value of the incident radiation coming from a single lamp can be obtained by using the E-VIE model and the usual methods that were described in sections VII, E,X, and XVII for direct and reflected radiation, respectively. For indirect radiation, only one reflection was considered; i.e., successive reflections after the first one were neglected. In addition, reflections that could be produced in reflectors others than the one specifically surrounding each lamp were also neglected. The radiation field was rigorously described taking into account also the following aspects: (i)the influence of the entrance and exit reaction wedges of the reactor on the overall production rate, was incorporated into the design equations by the proper redefinition of the limiting angles for 8; and (ii)in the case of reflected radiation, the restrictions imposed by the shape of the reflectors on the limiting angles for the variable 4 (Alfano et al., 1990). With the usual assumptions, the oxalic acid dimensionless mass balance can be written as follows:

The initial and boundary conditions are

YV,(O,y,P)= 1

(166.a) (166.b)

(166.c)

(166.d)

(166.e) An experimental verification of the model (Vicente et al., 1990) was performed by using the well-known photodecomposition of the oxalic acid and uranyl sulfate reacting mixture. The expression for the overall reaction rate was represented by eq 158. Thus, the reactor model was solved for a reacting solution with an initial concentration of 0.001 M uranyl sulfate and 0.005 M oxalic acid. Output conversions, as a function of the volumetric flow rate, using a two-dimensional mass balance model for the extreme values of /? = 0 and /? = n/4 were computed. Since differences were never larger than 2%, it was concluded that the additional numerical complexity required by the three-dimensional mass balance model is not justified. The reactor used four 15 W germicidal lamps. T h e reflector and lamp lengths were 34 cm and the reactor dimensions were 100 cm long and 1.85 cm in diameter. Figure 31 shows a compendium of all the theoretical predictions and experimental results. It must be stressed that the observed good agreement was obtained by using the model with the two-dimensional mass balance and a variable attenuation coefficient (a function of the oxalic acid concentration). The radiation balance was always three-dimensional.

x pred Figure 31. Experimental conversion vs predicted conversion. Open symbols, potassium ferrioxalate actinometer: (0)ellipse, (0) parabola, (v)direct radiation. Filled symbols, uranyl oxalate actinometer: (A) circumference. [Adapted from Cassano and Alfano (1991).]

Due to the important contributions of indirect radiation found in this system (60-70%), Esplugas et al. (1990) studied the effects of the shape and dimensions of the reflecting mirrors on the reactor performance. For a given geometrical arrangement, it can be shown that (i) there exists only one ellipse with one focus located at the center of the reactor and the other a t the center of the lamp, (ii)there exists only one parabola with its focus located at the center of the lamp, and (iii) there exist infinite circumferences, each one with a different radius. The flow of photons for each one of the geometries at issue was evaluated in order to characterize the incidence efficiency (Cerda et al., 1977). From the computed results, it was found that for a better use of the radiation emitted by the lamps, it is convenient t o employ elliptical reflectors. Besides, it was shown that reflectors of circular cross section may also reach a high efficiency provided that their location and dimensions are properly designed. In order to compare the incidence efficiency theoretical predictions with the experimental results, two multilamp reactors with elliptical (maximum incidence efficiency) and parabolic (minimum incidence efficiency) reflectors were built. As a comparison basis between both geometries, the reference radius (radius of intersection of the different reflectors) and the reactor and lamp dimensions were kept constant. Several actinometric verifications in a flow reactor were performed by using the potassium ferrioxalate reaction (Parker, 1953; Hatchard and Parker, 1956). The overall chemical reaction may be written as follows: 2Fe3+

+ CO, :-

+ hv

-

+

2Fe2+ 2C0,

Once more, the kinetics of this reaction can be described with a phenomenological approach as the one represented by eq 158; quantum yields were taken from the above-mentioned literature and conversions were expressed in terms of the ferrous ion. The experimental system had four 30 W germicidal lamps. The reactor and reflector lengths were 59 cm, and the lamp length was 81 cm. The radius of the reactor was 1 cm. The experiments were carried out using four elliptical or parabolic reflectors to collect and measure the effect of the total radiation (direct and


indirect) in each case, and with no reflecting mirrors to measure only direct radiation. Figure 31 also shows the comparison between model predictions and experimental results for elliptical and parabolic reflectors. It can be observed that in the two cases under study the agreement is very good. 4. Multitube-Single Lamp Reactor with Circular Reflector. Simple Reaction. This system is an arrangement that can be used to perform photochemical reactions under moderate pressures. Thus, n small diameter, tubular reactors are located around a tubular lamp and enclosed by a circular reflector (Figure 23). The system could be used, for example, for the photochlorination of methyl chloride in the liquid phase and at pressures between 450 and 1000 kPa (Cabrera et al., 1991c). This work was developed using the E-VIEmodel for the lamp. One must take into account the existence of direct and reflected radiation. The direct radiation can be determined with the usual procedures that were shown in sections M and X. The reflected radiation poses a problem of a higher degree of complexity than in the case of the elliptical reflector. Considering only a single reflection process, any reactor tube located in the annular arrangement can receive (i)direct radiation and (ii) indirect radiation from two different reflector regions, (ii.1)that part of the reflector that is closer t o the tube and (ii.2) the opposite part of the circular mirror. When the effects of the reflection produced by the reflector on one reactor tube are analyzed, an additional problem arises: the existence of all the other reactor tubes in the system produces a blocking process to many of the radiation ray trajectories. Thus, part of the radiation rays that could, otherwise, be reflected on the surface are obstructed. When this phenomenon is incorporated, it gives rise t o the appearance of umbra and penumbra zones on the reflector surface [for further details, refer to Tymoschuk et al. (1993a)l. The complete model (with direct and reflected radiation) was used only in the analysis of the incidence efficiency (ratio of the energy reaching the reactors and the energy emitted by the radiation source), but in any event, with an appropriate design the reflected radiation should be minimal; when this is the case, it can be neglected. (The “proper design” means t o place as many reactors as possible and to assign to each one of them the required optical path length to achieve complete absorption.) Fortunately, neglecting second order effects (such as the mutual interaction by reflection among reactors, for example) with some additional effort, results were extended to the multitubular system. Simulated results assuming a reactor with a constant absorption coefficient for the reacting mixture showed the existence of strong angular asymmetries due to the decisive participation of direct radiation in the total value of the radiation field. This situation translates to the mass and energy balances the unavoidable consideration of the angular coordinate to account for an angularly asymmetric reaction field. The design equations are similar t o the ones employed in reactors of circular cross section under laminar flow regime. The main difference appears in the mass balance because it must incorporate the angular component of the mass fluxes (see eqs 165-166.e). In this case the system was also operated under isothermal conditions. The reaction used to test the model was the potassium ferrioxalate photodecomposition previously mentioned

pred. Figure 32. Experimental conversion vs predicted conversion for potassium fenioxalate decomposition. Direct radiation: (A)m = 0.2 cm, ( 0 )m = 0.5 cm, (m) m = 1.0 cm. Total radiation: ( 0 )m = 0.5 cm. [Adapted from Cassano and Alfano (1991).1

for the multilamp reactor. The radiation field is a function of the actinometer concentration that changes along the reador length. Thus, there exists the coupling between the radiation attenuation and the reaction extent that the model must take into consideration. The experimental test in bench scale reactors used a 0.006 M solution of the ferric salt in three different reactors (0.2, 0.5, and 1.0 cm in diameter and 59.5 cm long). The radiation source was a germicidal U V arc lamp with 30 W of nominal input power (Tymoschuket al., 1993a,b). Figure 32 depicts the comparison between theoretical predictions and experimental results. As in previous examples, the agreement is very satisfactory. WL Application to a Pseudo-Homogeneous System 1. Gas-Liquid Tank Photoreactor Irradiated from the Bottom. Chlorination of Trichloroethylene and Pentachloroethane. This section presents a heterogeneous gas-liquid photochemical reaction carried out inside a perfectly stirred tank, semibatch reactor. The reactor was irradiated from the bottom by means of a tubular source located at the focal axis of a cylindrical reflector of parabolic cross section (Figure 21). The trichloroethylene and pentachloroethane photochlorination reactions were studied. The reactor had a continuous feed of pure or diluted (with nitrogen) chlorine gas that bubbled in a well-stirred batch of liquid reactant. Carbon tetrachloride was used as a diluent for the liquid phase. Unreacted gas was continuously eliminated from the reaction chamber (Alfano and Cassano, 1988a,b). Information about the reaction kinetics may be obtained from the data provided by the specific literature on gas and liquid phase chlorinations (Leermakers and Dickinson, 1932; Daiton et al., 1957; Walling, 1957; Poutsma and Hinman, 1964). The overall reaction may be represented by the following reactions:

C2HC1,

-C2HC1, -C2C1, +c1,

+c1,

where the first step is an addition reaction (fast) and the second a substitution reaction. The reaction mechanism including all possible steps,

Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 2191 Table 9. Definitions for the Dimensionless Mass Balances denomination symbol definition stable species concentration array intermediate species concentration array stable species reaction rate intermediate species reaction rate absorption rate

W?)

(vl,vYz,v3,v5,ydT

Y ( r )or Y(x,r)

(@1,@4,@6)T

@(x,t) O(x,t)

(?l,?2,?3,Q6,QdT (Ql,Q4,QdT

N(t)

(OJcL(Y~-Y2),0,0,O~T

is given by the following sequence:

initiation c1,

+ hv

-

propagation Cl'

+ C2HCl,

C2HC1,'

+ C1,

+ C,HCl, C2C1,' + C1,

Cl'

-

-

Q

2cl'

C,HCl,'

K,, K ,

+ Cl' C2C1,' + HCl C2C16+ Cl' C2HC1,

K,, k,, k4, k4, k,,

mean lifetime of both the stable and unstable species; so, both species are considered in a perfect mixing state. In the second, compared with the characteristic mixing time, the unstable species (of very short lifetime) are immobile; one can say that they are born, live, and die in the same place. Let us define the following notation: HC1, Clz, 2; czHC13, 3; CzHC15, 5 ; C2Cl6, 6; CI', 1;C2HCl4*,4; C2Cl$, 5; nitrogen is the inert diluent in the gas phase; C c 4 is the inert diluent in the liquid phase. Tables 9 and 10 give the definition array and the dimensionless mass balance equations in the liquid bulk for the models generated by considering mixing cases 1 and 2. A system of integro-differential equations with specified initial values was obtained. In addition, expressions for the absorption rate N(z) and the reaction rates Q(x,z)must be found to complete the mathematical description of the process. Obviously,the knowledge of the reaction rates requires the proper description of the radiation field inside the reactor. The vector N(z) represents the absorption rate into the liquid bulk of reactants. If one considers that in the liquid phase only gaseous chlorine is absorbed, the dimensionless expressions for each component of the vector are given by

r;

N J z )= 0 (i f 2)

(167)

N,(z>= K,[Y~,(T)- Y,(Z)I

(168)

Ku

termination where C2HC1,' C,Cl,'

+ C2HC1,' + C2C1,'

-

+ C2HC1,' Cl' + C2C15' C2HC1,' + C2C1,' Cl'

-

products

products

.

products

Table 10. Dimensionless Mass Balance Eauations mixing states model model 2

total time of reaction absorption time

-[

]

(169)

ci,

chlorine in gas-liquid interface] (170)

products

stable species intermediate species stable species intermediate species

tT

- l/k,A,

Gef

The analysis of the problem must take into account the following aspects: (i) the mixing states of the reacting species, (ii) the gas absorption rate, (iii) the reaction rate, and (iv) the radiant energy distribution inside the reactor. The existence of a nonuniform radiation field produces nonuniform reaction rates. Hence, even if one assumes perfect mixing for reactants and stable products, concentration gradients of free radicals and atomic chlorine may be present. To consider these mixing states, a comparative analysis between the mean lifetime of all reacting species and a characteristic mixing time was performed. Mainly, two realistic mixing conditions may be considered: (i) mixing state 1, perfect mixing for stable species, free radicals and atomic chlorine; and (ii) mixing state 2, perfect mixing for stable species and no mixing for unstable intermediate species. In the first case the mixing characteristic time is less than the

model 1

--_

y1--- [dimensionless concn of 2-

products

-

K

k,

perfect mixing perfect mixing perfect mixing no mixing

Fast reaction rates may affect the gas absorption rates. We know that photochemical reactions have an initiation step that is always spatially dependent because the radiation field is intrinsically nonuniform. Under these circumstances if an enhancement factor different from 1 is present, since the reaction rate is spatially dependent, the mass transfer coefficient (modified by the chemical reaction) will also be a function of position. If this is the case, modeling of the mass transfer rate for the photoreactor will be a very difficult task. In this case it was shown that the reaction in the liquid phase is slow enough t o be negligible during the lifetime of the liquid surface elements (tDiff .Y(x,t) propagation rate

+

In eqs 180 and 181since stable species are well mixed, a(z)and /3(dare not a function of position. Conversely, the values given by eqs 177, 178, and 179 must be averaged over the reactor volume:

R(x,z).FY(x,t) termination rate 2g(x,t) = 0 (175) initiation rate

(183)

where the definitions given by Table 11were employed. Note that each term is a function of position and time. Applying the long chain approximation (Gavalas, 1966): A(z>.Y(x,d= 0

(176)

From eq 176 it is possible to get Y4(x,z) and Y5(x,z) and then, introducing these concentrations into eq 175,

For the stable species one can write


(NX,~)) = B(t>.(Y(x,t)) + (g(x,t)>- (t(x,t)) (185) where (Y(x,z)) is given by eqs 177-184, and the other variables may be calculated from the available kinetic information and the predictions of the radiation field. Similar equations can be derived for model 1. The only differences are (i) eqs 177 and 178 do not require the averaging procedure because all concentrations are independent of position, (ii)when calculating the concentration of W(z) one gets the “square root of the average rate of initiation” instead of the “average of the square root of the rate of initiation”, and (iii) in the equivalent equation t o eq 185 the only term that must be averaged over the reactor volume is g(x,z). To complete the theoretical description of the models, it is necessary to introduce the radiation field expression in the initiation reaction array and in the intermediate species concentration array. The radiant field was evaluated by means of a proposed model that had been tested experimentally in previous instances (Alfano et al., 1985, 1986c,d). A nonfluorescent, arc lamp was employed, and the E-VIE model was applied. Direct radiation was modeled by applying the equations preand X. They are essentially sented in sections VII, E, the same than the ones derived for the annular reactor. Reflected radiation was modeled as described in section XVII; the resulting equations, applied to the cylindrical reflector of parabolic cross section can be found in Alfano and Cassano (1988a,b). The results are expressed in from which the terms of the incident radiation (G,) monochromatic LVREA can be immediately obtained. Extension to the polychromatic irradiation was done by the same methods described before (section XVI). The reader may refer t o these papers for further details. To complete the evaluation of the LVREA, we must consider that in theory the reacting system is heterogeneous (gas-liquid). It has been said in section XI1 that a proposed approximation considers the system as pseudo-homogeneous using an effective absorption coefficient. Thus, the strategy for the modeling problem is to make use of the LVREA expressions for homogeneous media but incorporating a semiempirical equation that has been developed to account for changes in the radiation field produced by the presence of gas bubbles. This approach was adopted for this case and the Yokota et al. (1981)correlation was used (eq 103). This effective coefficient was correlated as a function of the optical properties of the liquid phase, the gas holdup, and the specific interfacial area. Finally, it should be mentioned that in certain cases it may be necessary to put a protecting cover over the emission system that can also be used to support the cylindrical reactor (Alfano and Cassano, 1988a). If this is the case, some parts of the reactor will receive radiation from the whole lamp and reflector, while other parts of the reaction space will receive only partial irradiation (due to the obstruction of the cover). Consequently, for each incidence point inside the reactor we have to determine which portion of the parabolic reflector or source is capable of illuminating such a point. From the mathematical point of view, the integration interval (limits, sections X and XVII) for each point of incidence must be adjusted. The circular hole of the lamp-reflector cover system defines for each point of incidence a set of limiting angles (ec,&I that must be calculated and afterward compared with the limiting angles defined for the integration of the source volume. For a more detailed information the reader is referred to Alfano et al. (1985, 1986c,d).

1.00

d

x”

0.50

0.00 0 .oo

0.50

I .oo

Xpred

Figure 33. Experimental conversion vs predicted conversion for trichloroethylene and pentachloroethane photochlorination. (0) Addition reaction, (A)substitution reaction for quartz bottom plate, (A)substitution reaction for pyrex bottom plate. [Adapted from Cassano and Alfano (19911.1

The experimental verification of the model was carried out in a 2 L stirred tank photoreactor built of F’yrex glass with a quartz bottom to provide a wall transparent to the entrance of the radiant energy. A 360 W arc lamp producing polychromatic light was employed. Each experimental run demanded about 6 h, of which the first four were carried out with nitrogen to eliminate impurities (mainly oxygen and moisture) and to reach the steady state operation of the lamp emission; afterward, flow rate adjustment of the streams of chlorine and nitrogen was effected. Further details regarding the device and experimental procedures may be found in Alfano and Cassano (1988b). Afterward, we compared the experimental results with theoretical predictions from both models and found an acceptable coincidence when considering the unstable intermediate species in a nonmixing state (model 2). Figure 33 summarizes the main results obtained by applying model 2. The existence of a diffusional subregime for the addition reaction (fast) was experimentally verified. The process rate (photoreactor performance) was found t o be independent of the reaction kinetics and consequently of the radiation distribution inside the reactor. The process rate increased with increasing stirring speed. The chlorination of pentachloroethane (slower substitution reaction) was also experimentally studied, and it was found that it proceeds under the kinetic subregime; the performance was sensitive to changes in the reactor bottom plate (from quartz to Pyrex glass for example) and experiments agreed well with model predictions. Hence, the photoreactor performance was kinetically dependent and an accurate knowledge of the radiation field was indispensable. Both types of reaction subregimes have been previously described by Astarita et al. (1983). Using these differences in both reaction rates, it was found that controlling the operating conditions it is possible to achieve almost 100% selectivity in the transformation of trichloroethylene into pentachloroethane .

XX. Applications to Heterogeneous Systems At this point we must depart from the format used in the preceding sections of this paper. We have indicated that methods, results, and conclusions will be

2194 Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 Table 12. Main Contributions of Santarelli and Co-Workers RTE geometry type of reaction method of solution summary plane slab photosensitized Fourier integral effects of scattering and mixing on conversion transform projection procedure influence of a reflecting boundary on radiant energy absorption non-photosensitized projection procedure effects of scattering and boundary reflectivity on conversion analysis of sun-driven photochemical reaction derivation of RTE; review of solution methods for RTE derivation and discussion of RTE; application to photochemical reaction effects of scattering and mixing on conversion annular photosensitized Monte Carlo of continuous reactor evaluation of 2-D radiation field within photocatalytic Monte Carlo photochemical reactor

r =0

I

pL’0

T =To

( r - z plane)

authors (year) Stramigioli et al. (1978) Spadoni et al. (1980) Santarelli et al. (1982) Stramigioli et al. (1982) Santarelli (1983) Santarelli (1985) Spadoni et al. (1978) Pasquali et al. (1993)

by the same group, have been the existing knowledge. Their most important features will be described in what follows. Table 12 provides a summary of the main achievements. An attempt of classification has been made along the following lines: (i) the employed geometry (plane slab and annular), (ii) the type of chemical reaction considered in the analysis (photosensitized, nonphotosensitized, and photocatalytic), and (iii) the mathematical method of solution employed to solve the radiative transfer equation (semianalytical methods and Monte Carlo simulation). 1. Plane Slab Geometry. Several papers of Santarelli and co-workers have analyzed the heterogeneous system performance using the plane slab geometry for the model regardless of the actual shape of the reactor. This simple geometry-in some cases an extreme idealization-permitted the study of the basic aspects of the problem. Normally, conclusions obtained in this way can be extrapolated-at least qualitatively-to more complex situations. However, in same cases this geometry also provides the appropriate representation of the real problem, for instance, in applications related to the use of solar energy for algae growth or t o the storage of energy in solar ponds. The representation of the system was made in a onedimensional Cartesian coordinate frame (Figure 34a) considering a plate of thickness L where absorption and scattering of radiation takes place; monochromatic radiation, no emission, azimuthal symmetry, and isotropic scattering (p = 1)were assumed. The radiative transfer equation (RTE) and the corresponding boundary conditions are

Figure 34. Coordinate systems for radiative transfer in heterogeneous media. (a) Plane slab geometry, (b)annular geometry.

described in this Review when they are based on the fundamental principles of radiation transport and chemical reaction engineering science. In this respect we must say that our knowledge base on heterogeneous photoreactor analysis and design is very limited. The most significant contributions-mainly produced with theoretical (either analytical or computational) results-are due to the Santarelli’s group. Just recently, some additional work, involving theory and experiments has been reported and it will be also described. Back in 1978, Stramigioli et al. and Spadoni et al. established a sound theoretical frame for analyzing radiation transport in participating and reacting heterogeneous media. During several years these, and subsequent publications

I,(To,-p) = @y”(t,,p)+

2d!L1Iv(zo,p’)p’dp’(p > 0) (188) with p = cos 8 5=p,x

(189) (190)

Equation 187 provides a generic form of the angular dependence of the boundary condition, in order to include the different cases that were studied (diffuse

Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 2195 emission, emission in parallel planes normal to the surface of radiation entrance, direct component of solar radiation, etc.). Equation 188 permits description of the general case of an opposite boundary having specular or diffise types of reflection. Notice that in these equations one deals with one spatial variable (z or x ) and one directional variable (u or 8). Photosensitized reactions were investigated. From the solution of the RTE point of view, since the absorption coefficient of the participating medium remains constant, one has a radiation field that is independent of the progress of the reaction inside the reactor; i.e., there is no coupling of the RTE with the mass balance of the radiation-absorbing species. Then, the solution of the different equations for modeling the reactor (e.g., mass and energy balances) is simplified. Nonphotosensitized reactions were also studied, analyzing the case where the radiation-absorbing species also reacts and changes its composition; i.e., the radiation field changes with the progress of the reaction. Different solution methods were employed. This is not a trivial part of the problem. Solution techniques for solving the RTE are not commonly used in chemical engineering problems. A revision on rigorous and approximate methods of solution has been presented by Ozisik (1973) and Santarelli (1983). All these methods were originally developed to investigate neutron transport in nuclear engineering problems; this transport is also governed by equations having very similar integrodifferential mathematical structure. For the plane slab geometry, Santarelli and co-workershave used what are called rigorous or semianalytical procedures. The use of different solution techniques can be found in Boffi et al. (1977, 1979), Spiga et al. (1980), and Santarelli et al. (1980). Working with photosensitized reactions, Stramigioli et al. (1978) reported a study of a plane slab of finite thickness irradiated with a monochromatic and parallel beam of radiation. Scattering effects on the reactor conversion were investigated for two extreme conditions of mixing: perfect mixing and no mixing. Similarly, Spadoni et al. (1980) modeled a plane slab having one boundary with diffuse emission and the opposite side with a surface that produces diffuse or specular reflection to the arriving radiation. The authors investigated the effects produced by the different reflecting surfaces on the radiation absorption inside the reactor. For nonsensitized reactions, Santarelli et al. (1982) studied a plane slab reactor having one boundary with a diffise emission of constant intensity and the opposite boundary with diffuse reflection. The analysis was centered in the effects of scattering, initial optical thickness, and boundary reflectivity on the conversion of a batch reactor. Stramigioli et al. (1982) used the same geometry but for a reactor irradiated by parallel beams of radiation having a direction that changes with time (simulating the direct component of solar irradiation). The model was applied t o a batch reactor. In 1983 and 1985 Santarelli published two review papers on radiative transfer in plane, participating and reacting media. The basic equations of radiative transfer were presented and the solution of the RTE coupled with the mass balance for a simple reaction was discussed. The first paper (1983) also presents a summary of the more important methods of solution for the RTE; a comparison of the obtained results (in terms of different radiation field variables) using some of those methods was also made.

2. Annular Geometry. Annular geometry constitutes the most common type of reactor employed in photochemical processes. Usually it is easier to construct and operate; additionally, the energy coming out from the lamp can be used with very high efficiency. Conversely, modeling of this reactor when absorption and scattering are present is not simple. Although the assumption was made that there were no surfaces with significant reflection to account for, the solution of the RTE with scattering in cylindrical geometry, even under the assumption of spatial angular symmetry, requires consideration of two spatial variables (r and z ) and two directional variables (8 and 4) (Figure 34b). Considering monochromatic radiation, with no emission, the RTE and associated boundary conditions, in cylindrical geometry, can be written as

Boundary condition 192 represents radiation entering the reactor at the inner wall; it is described by a generic function f(z,O,#). For example, the lamp radiation emission model (section E)can be introduced here. Equation 193 represents the condition at the outer reactor wall, indicating that it is transparent to radiation; i.e., there is no reflection. Initially, Spadoni et al. (1978) studied photosensitized reactions mainly motivated by situations encountered in photoinitiated polymerizations and in the presence of suspended, noncatalytic solid particles in some water decontamination processes. Later on, Pasquali et al. (1993) worked on photocatalytic reactions in view of the increasing interest raised by advanced oxidation processes for water and air purification using titanium oxide and U V radiation. Strictly speaking, in theory, from the RTE point of view, both systems are equivalent, the key point being the existing uncoupling of the radiation equation from the mass balance; then, the LVREA is a function of the spatial variables and is independent of the reaction progress. Semianalytical techniques of solution employed in plane slab geometry are very convenient, particularly if isotropic scattering is assumed. In multidimensional geometries, using curvilinear coordinates, these methods are either too complicated or cannot be applied at all. Then, it is of no surprise that in order t o tackle the annular geometry problems, other methods had been explored. Santarelli and co-workers have used Monte Carlo simulation; instead of computing the radiation intensity by application of eqs 191-193 and from these results calculating the LVREA, they have simulated the behavior of a statistically significant number of photons. The life of these photons is followed from their “birth” at the radiation source t o their “fate” by absorption or by scattering out of the reactor boundaries. Spadoni et al. (1978) is surely one of the first (if not the first) works where a cylindrical (annular) heterogeneous photoreactor is rigorously modeled. Isotropic


scattering was assumed. The paper deals with a reactor operated in a continuous fashion where a photosensitized reaction is produced. Radiation and species concentration profiles were computed, and scattering as well as reaction space optical thickness effects on the reactor performance were analyzed for conditions of perfect mixing and no mixing. Pasquali et al. (1993) studied a photocatalytic reaction, relaxing the restriction of isotropy for the existing scattering. Radial profiles of the LVREA were calculated as a function of the following characteristics of the system: optical thickness, scattering albedo, and phase function. All these contributions have generated a good theoretical basis for studying heterogeneous photoreactors. Unfortunately, almost no experiments were performed and system parameters used for presenting the model predictions (absorption and scattering coefficients for example) are usually the result of estimations within plausible limits but with no correspondence with any specific system. Results and conclusions, which are good indeed, can be used only on a qualitative basis. In spite of it, several important conclusions extracted from Santarelli and co-workers group can be summarized as follows: 1. Compared with an equivalent homogeneous case, scattering produces a decrease in the amount of radiation absorbed by the system. This is due to the escape of photons out of the system boundaries provoked by the scattering centers. 2. When the optical thickness for absorption increases, the probability of escape by photons decreases (or, equivalently, the probability of photon absorption increases). As a consequence, scattering effects are less significant in reacting media of very high optical thickness. 3. In a perfectly mixed reactor, scattering always produces a decrease in the rate of a photosensitized reaction. Conversion depends upon the absorbed radiation; since it is decreased by scattering, the reaction rate also decreases. 4. In an optically thick reacting system that is operated under poor mixing conditions (and with a photosensitized reaction), scattering can produce an increase in the reactor yield. Under these conditions scattering produces a redistribution of photons in space and, consequently, of the LVREA. The new photon distribution diminishes the typical radiation absorption nonuniformities that result from radiation attenuation in highly absorbing media and helps to compensate the reduction in yield resulting from imperfect mixing in the reactor. This positive effect on conversion is able to counteract, and sometimes overcome, the escape of photons due to scattering. 5. When a photosensitized reaction is conducted in a medium with scattering, for intermediate values of the optical thickness, reactor walls with reflecting surfaces may produce an increase in the rate of radiation absorption and, consequently, in the reactor yield. 3. Additional Work. Very recently, in a series of papers, Alfano et al. (1994, 1995) and Cabrera et al. (1994, 1995) have analyzed both theoretically and experimentally scattering effects in photoreactors. To evaluate the radiation intensity of the photoreactor system, a planar geometry was adopted. The photoreactor was irradiated by a tubular UV source with a parabolic reflector (Figure 35). The model considered that radiation propagates in only one dimension and only one angular variable for scattering was included under the assumption of azimutal symmetry. The local

A

OUTLET

GLASS WINDOW

n

REACTOR/

n

INLET

Figure 35. Planar photoreactor irradiated by a tubular U V source with a parabolic reflector. [Adapted from Alfano et al. (1994a).l

volumetric rate of energy absorption (LVREA) that is needed to write the rate of reaction was obtained from the radiation intensities &r solving the RTE. Solution was obtained by application of a numerical technique known as the ordinate discrete method (Duderstadt and Martin, 1979). The optical parameters (absorption and scattering coefficients) for solving the radiative transfer equation were measured and Werent phase functions-according to the type of particles employed-were adopted. Bench scale verifications were carried out by studying the effects produced by addition of scattering centers (inert, transparent, spherical, silica beads) to a liquid solution where, upon radiation absorption, the homogeneous photodecomposition of the uranyl oxalate complex was measured. In this case, a specular, partial reflection phase function model was used (Siegel and Howell, 1992). The experimental device was an isothermal, well-stirred, continuous reactor operated inside a batch system with recycle. The complete description of the system included (1) a rigorous model of emission for the tubular lampparabolic reflector system derived from previous work published by Alfano et al. (1985, 1986c,d), (2) a model of the radiation field distribution that exists at the interior face (a side made of ground glass) of the surface of radiation entrance t o the reactor, (3) a very simple model for the recycling reactor system, (4) an independent experimental determination of the radiation absorption coefficient (for the homogeneous phase) and the scattering coefficient (for the suspension of micron size, transparent, spherical particles in water), and (5) a model derived from the geometric optics (Siegel and Howell, 1992) for the scattered radiation distribution (the phase function) that was assumed to respond to a specular, partial reflection process. Figure 36 shows the results of the experimental verification of the model. Afterward, the scattering phenomenon inside the heterogeneous reactor was studied in detail, analyzing the effects produced on the system by changes in its more important parameters. Using the inert scattering center concentration and the radiation-absorbing species concentration as independent variables, the resulting consequences on the following aspects were described: (i) angular distribution of radiation intensities inside the reactor, (ii) spatial distribution of the LVREA inside the reactor, (iii) ratio of the reactor volume averaged reaction rates with and without scattering, and (iv) radiation energy absorption efficiency inside the reactor. Also, changes in the reactor performance predictions were discussed when different forms (models) of the scattering phase function were used to solve the radiative transfer equation. The validated model was also used to study scattering effects by radiation-absorbing particles. Titanium oxide

Ind. Eng. Chem. Res., Vol. 34,No. 7, 1995 2197

(R,,)pred x IO9(mole cm-3 s-’ Figure 36. Experimental average reaction rate vs predicted average reaction rate for uranyl oxalate decomposition in heteroC,, = 0 g ~ m - (~A ), Cmp = 3 x g cm-3, geneous media. (0) g~ m - ~ . (A)C,, = 6 x

suspensions were studied. The absorption and scattering coefficients were spectrophotometrically measured and a diffise reflection phase function model was employed (Siege1 and Howell, 1992). The model was applied to two different problems: (i) to evaluate the LVREA in a system without chemical reaction and (ii) to evaluate the overall quantum yield in the photocatalytic decomposition of trichloroethylene. So far, predictions from the model and experimental results permit the following conclusions: 1. Addition of scattering centers to a very well mixed photochemical system always produces a diminishing in the average reaction rate with respect t o the case where there is no scattering. The escape of photons is important a t the surface of radiation entrance due to the significant contribution of back-scattering. 2. When the system has very low radiation absorption, addition of inert particles (scattering centers) does not produce a significant decrease in the reaction rate. When absorption is higher, up to an intermediate optical thickness, the reaction rate always decreases. When absorption is very high, the scattering effect is again less significant. There exists an ample range of concentrations where one observes that the effect takes its maximum value. 3. Addition of scattering centers increases the optical thickness of the reactor and produces steeper profiles of the LVREA when it is represented as a function of the spatial variable ( x ) . The practical consequence is an increase in the reaction rate in regions closer to the reactor surface of radiation entrance and lower rates in the other parts of the reactor. These changes are more significant at low and medium values of the radiation-absorbing species concentration. 4. The concept of absorption efficiency (absorbed radiation energy/available radiation energy at the reactor boundary) was used to study these effects also in a quantitative manner. This efficiency is always less than or equal to 1 regardless of the reactor and operating conditions. It was found that the absorption efficiency in a heterogeneous reactor is always smaller than one and, for well mixed systems, it is always smaller than that of the equivalent homogeneous reactor (without scattering centers). 5. The absorption efficiency was also used to study the effects on the prediction produced by the adoption

of different models for the phase function (distribution function for scattering). Results for the cases in which isotropic and diffise reflection phase function models were used do not show important differences. Conversely, the case of a specular, partial reflection phase function model differs in a significant way with the other two; at the same time, its results show the smallest differences when they are compared with the purely homogeneous case. 6. Under normal conditions, scattering effects produced by titanium dioxide water suspensions in photocatalytic reactors are very significant. For titanium oxide suspensions it was found that an accurate knowledge of the absorption and scattering coefficients, as well as a good approximation to the phase function description, are indispensable to compute the proper value of the volume-averaged LVREA. 7. To compute the LVREA in titanium dioxide suspensions, methods that do not include the proper solution of the RTE-even in its simplest form (i.e., onedimensional models)-produce sigmficant errors. Among them, the most widely employed approach (a method using homogeneous actinometry) renders results that, for the explored operating conditions, may be off by an error larger than 500%, although greater deviations are likely to occur. 8. Results also indicate that the use of radiation flux measurements a t the reactor entrance or homogeneous actinometry inside the reactor volume for evaluating the energy yield in photocatalytic reactions may lead t o reporting very low quantum yields. The reason is that important scattering effects are not accounted for and the LVREA ie overestimated. 9. It was found that decreasing the particle size at constant catalyst concentration increases the radiationabsorption aptitudes of the solid suspension. This effect is more significant at low particle concentrations.

XXI. Conclusions Through one detailed example and references to several different applications, it has been shown that there exist photoreactor design methods for homogeneous systems that allow a correct a priori prediction of the reactor performance or permit its precise design. These methods do not include adjustable parameters. Hence, they can be used equally well for any reactor size. If the mass and thermal fluxes are predictable, and the intrinsic reaction kinetics is known, the proposed methodology produces trustworthy results. It seems clear that a similar effort should be exercised in order to develop an equivalent state of the art in heterogeneous photoreactors. As far as reactor design is concerned, the research lines that should be explored deeply, resorting to fundamental principles, are the following: a. Experimental techniques to measure the optical properties of different photocatalytic systems (solid suspensions and fured bed reactors) particularly in the U V range should be developed. b. Simplified kinetic schemes suitable for representing the very complex mechanisms of photocatalytic reactions should be developed. This is particularly important because the most promising applications seem t o be oriented toward air and water pollution abatement. In these cases two main difficulties can be foreseen: (i)the reaction environment (e.g., pH) changes with the reaction extent and (ii) the reaction mixture

2198 Ind. Eng. Chem. Res., Vol. 34,No. 7,1995

is quite complex (several pollutants and impurities) and the initial composition may change with time. Both circumstances may render impossible the utilization of too sophisticated kinetic models. c. Another area for development is enlargement of the knowledge base of intrinsic reaction kinetics data, particularly for complex mixtures. d. Additionally, development of more efficient catalysts of proven mechanical and chemical stability, and improvement of catalyst immobilization methods is needed. e. Solution of the design equations in reacting media when absorption and scattering are present should be accomplished. Emphasis must be placed in problems involving bidimensional geometries having a radiation intensity distribution that is a function of more than one directional variable. f. Development of design methods for fixed-bed photocatalytic reactors should be explored. From the economical point of view, in order to avoid downstream separation process costs associated with the suspended photocatalyst, this type of reactor seems to have a better prospective. g. Development of precise models for solar irradiated reactors which could be an additional attraction for the process economy should also be explored.

k = kinetic constant, m3mol-' s-l; also empirical coefficient used in eq 102 kL = mass transfer coefficient, m s-l L = length, m N = absorption rate, dimensionless Nu = hcr& Nusselt number, dimensionless nR = photon density number, photon m-3 sr-l n = unit normal vector, dimensionless P = pressure, Pa; also radiant power, einstein s-l Pe = m(u+)/Dlm, mass Peclet number, dimensionless PeT = emuCp~(uz)/A, thermal Peclet number, dimensionless p = phase function, dimensionless Q = heat of reaction defined by eq 14,dimensionless R = reaction rate, mol s-l m4 RL = lamp radius, m r = radius, m; also radial coordinate, m S, = selectivity of product i, dimensionless s = linear coordinate along the direction Q, m T = temperature, K t = time, s U = velocity defined by eq 3, dimensionless V = volume, m3 u = velocity, m s-l W = gain or loss of energy, einstein s-l m-3 sr-l X,= conversion of species i, dimensionless x = position vector, m x = size parameter, dimensionless x , y, z = rectangular Cartesian coordinate, m

Acknowledgment

Greek Letters a = molar Naperian absorptivity, m2 mol-' 6 , = volumetric extinction coefficient, m-l; also cylindrical coordinate, rad y = ria, radial coordinate, dimensionless r = reflection coefficient, dimensionless Q = unit vector, dimensionless EG = gas holdup, dimensionless 5 = dLR,axial coordinate, dimensionless t? = linear coordinate defined by eq 121,dimensionless 8 = spherical coordinate, rad K = volumetric absorption coefficient, m-l A = absorption number defined by eq 120,dimensionless; also clearance, m A = wavelength, m; also thermal conductivity, W m-l K-I p = viscosity, g m-l s-l; also the quantity cos 8, dimensionless v = frequency, s-l v, =j component stoichiometric coefficient e = hemispherical reflectivity, dimensionless; also spherical coordinate, m emlx= density of the reacting mixture, kg m-3 u = volumetric scattering coefficient, m-' t = temperature defined by eq 15, dimensionless; also variable defined in eq 190 Y = transmission coefficient, dimensionless Q = quantum yield, mol einstein-l Qoverall = overall quantum yield defined in eq 25, mol

The authors are grateful to Consejo Nacional de Investigaciones Cientificas y TBcnicas (CONICETI and to Universidad Nacional del Litoral (U.N.L.) for their support to produce this work. Special mention has to be made to Research Trends in Photochemistry and Photobiology and Research Trends in Chemical Engineering for permitting partial reproduction ofpublished material.

Nomenclature A = area, m2;also preexponential factor, m3mol-' s-l (with the exception of CP CP homogeneoustermination step, where the units are m6 moP2 s-l) A, = specific surface area, m-l C = molar concentration, mol m-3 Ci = concentration of the i component, mol m-3 ,C , = particle mass concentration, kg m-3 C, = heat capacity, J mol-' K-' c = speed of light, m s-l AH = heat of reaction, J mol-' D = diffusion coefficient, m2 s-l db = bubble diameter, m d, = particle diameter, m E = activation energy, J mol-' e* = local volumetric rate of radiant energy absorption, einstein m-3 s-' F = feed molar ratio (chlorindmethane), dimensionless fR = photon distribution function, dimensionless fv = particle volume fraction, dimensionless G = incident radiation, einstein s-l m-2 Ge = rR/LR, geometric number, dimensionless g = gravitational acceleration, m s-2 h = Planck's constant, J s; also heat transfer film coefficient, W m-2 K-l I = specific intensity, einstein s-l m-2 sr-l J = initiation number defined in eq 123,dimensionless j . = energy emission, einstein s-1 m-3 sr-1 K = kinetic constant, dimensionless KL = mass transfer coefficient, dimensionless

+

C#J

x

einstein= spherical coordinate, rad

= inner radius-outer radius ratio, defined by eq 2,

dimensionless Y, = CJC",, concentration, dimensionless Q = unit vector in the direction of propagation, dimension-

less Q = solid angle, sr

a, = RJR/(u,)G, reaction rate, dimensionless Qw,L = Rw,,/(ur)G, reaction

rate at the wall, dimensionless

w = scattering albedo, dimensionless

Subscripts 0 = inlet condition; also relative to the surface of radiation

entrance

Ind. Eng. Chem. Res., Vol. 34,No. 7, 1995 2199 1 = integration lower limit 2 = integration upper limit

b = relative to blackbody radiation C = relative to the heat exchange fluid c = characteristic variable E = property of a ray emerging from the source Exp. = experimental value e = exit eff = effective property F = relative to a fured Cartesian coordinate system G = global reaction I = incident point or an incident ray property i = relative to species i i = inlet init = relative to the initiation step j = relative to speciesj k = relative to species k L = relative to the lamp LR = reactor length condition Mod. = model value n = relative to the normal overall = overall reaction ox = relative to oxalic acid P = point of reflection p = particle property prim = primary reaction R = relative to the reactor Rf = reflected radiation property Ri = relative to the inner wall of the reactor R, = relative to the outer wall of the reactor reac = characteristic reaction time resid = mean residence time s = relative to the radiation source z = relative to the z-axis 1 = dependence on wavelength v = dependence on frequency f2 = relative to the direction of propagation Superscripts ’ = value projected on the x-y plane; also denotes a reverse reaction (k)= relative to the overall reaction k 0 = relative to the surface of radiation entrance a = relative to absorption e = relative to emission exp = exponential form lin = linear form = initial conditions s-in = relative to in-scattering s-out = relative to out-scattering O

Special Symbols ( ) = average value [=I = “has unit of’ A = atomic or a free radical species

Literature Cited Alfano, 0. M.; Cassano, A. E. Modelling of a Gas-Liquid Tank Photoreactor Irradiated from the Bottom I. Theory. Znd. Eng. Chem. Res. 1988a,27, 1087-1095. Alfano, 0.M.; Cassano, A. E. Modelling of a Gas-Liquid Tank Photoreactor Irradiated from the Bottom. 11. Experiments. Znd. Eng. Chem. Res. 1988b,27, 1095-1103. Alfano, 0. M.; Romero, R. L.; Cassano, A. E. A Cylindrical Photoreactor Irradiated From the Bottom. I. Radiation Flux Density Generated by a Tubular Source and a Parabolic Reflector. Chem. Eng. Sci. 1985,40,2119-2127. Alfano, 0.M.; Romero, R. L.; Cassano, A. E. Radiation Field Modelling in Photoreactors. I. Homogeneous media. Chem. Eng. Sci. 1986a,41,421-444. Alfano, 0.M.; Romero, R. L.; Cassano, A. E. Radiation Field Modelling in Photoreactors. 11. Heterogeneous Media. Chem. Eng. Sci. 1986b,41, 1137-1153.

Alfano, 0. M.; Romero, R. L.; Cassano, A. E. A Cylindrical Photoreactor Irradiated from the Bottom. 11. Models for the Local Volumetric Rate of Energy Absorption with Polycromatic Radiation and their Evaluation. Chem. Eng. Sci. 1986c,41, 1155-1161. Alfano, 0.M.; Romero, R. L.; Negro, C. A.; Cassano, A. E. A Cylindrical Photoreactor Irradiated from the Bottom. 111. Measurement of Absolute Values of the Local Volumetric Rate of Energy Absorption. Experiments with Polychromatic Radiation. Chem. Eng. Sci. 1986d,41,1163-1169. Alfano, 0.M.; Vicente, M.; Esplugas, S.;Cassano, A. E. Radiation Field Inside a Tubular Multilamp Reactor for Water and Waste Water Treatment. Znd. Eng. Chem. Res. 1990,29,1270-1278. Alfano, 0.M.; Cabrera, M. I.; Cassano, A. E. Modeling of Light Scattering in Photochemical Reactors. Chem. Eng. Sci. 1994, 49,5327-5346. Alfano, 0. M.; Negro, A. C.; Cabrera, M. I.; Cassano, A. E. Scattering Effects Produced by Inert Particles in Photochemical Reactors. 1. Model and Experimental Verification. Znd. Eng. Chem. Res. 1995,34,488-499. Astarita, G. Mass Transfer with Chemical Reactions; Elsevier: Amsterdam, 1967. Astarita, G.; Savage, D. W.; Bisio, A. Gas Treating with Chemical Solvent; Wiley: New York, 1983. Ayscough, P. B.; Cocker, A. J.; Dainton, F. S.; Hirst, S. Photochlorination Studies. 6. cis-l,2 Dichloroethylene. Trans. Faraday SOC. 1962,58, 295-307. Baginski, F. C. Liquid-Phase Photochemical Reactions in a Flow System. D. Eng. Dissertation, Yale University at New Haven, CT, 1951. BofTi, V. C.; Santarelli, F.; Stramigioli, C. Rigorous Constructive Solutions in Radiative Heat Transfer. Znt. J . Eng. Sci. 1977, 15,607-621. Boffi, V. C.; Santarelli, F.; Spiga, G.; Stramigioli, C. Radiative Transfer in an Absorbing Scattering Slab Bounded by Emitting and Reflecting Surfaces. Int. J. Heat Mass Transfer 1979,22, 1705-1717. Braun, A.; Maurette, M. T.; Oliveros, E. Technologie Photochemique, 1st ed.; Presses Polytechniques Romandes: Lausanne, 1986. Cabrera, M. I.; Alfano, 0. M.; Cassano, A. E. Selectivity Studies in the Photochlorination of Methane. I. Reactor Model and Kinetic Studies in a Non-isothermal, Polychromatic Environment. Chem. Eng. Commun. 1991a,107,95-122. Cabrera, M. I.; Alfano, 0. M.; Cassano, A. E. Selectivity Studies in the Photochlorination of Methane. 11. Effect of the Radiation Source Output Power and Output Spectral Distribution. Chem. Eng. Commun. 1991b,107,123-150. Cabrera, M. I.; Alfano, 0. M.; Cassano, A. E. Nonisothermal Photoclorination of Methyl Chloride in the Liquid Phase. AZChE J. 1991~, 37,1471-1484. Cabrera, M. I.; Alfano, 0. M.; Cassano, A. E. Novel Reactor for Photocatalytic Kinetic Studies. Znd. Eng. Chem. Res. 1994,33, 3031-3042. Cabrera, M. I.; Alfano, 0. M.; Cassano, A. E. Scattering Effects Produced by Inert Particles in Photochemical Reactors. 2. A Parametric Study. Znd. Eng. Chem. Res. 1995,34,500-509. Calvert, J. G.; Pitts, N. R. Photochemistry; Wiley: New York, 1966. Cassano, A. E.; Silveston, P. L.; Smith, J. M. Photochemical Reaction Engineering. Znd. Eng. Chem. 1967,59,18-38. Cassano, A. E.; Alfano, 0. M.; Romero, R. L. Photoreactor Engineering: Analysis and Design. In Concepts and Design of Chemical Reactors; Whitaker, S., Cassano A. E., Eds.; Gordon and Breach: New York, 1986; pp 339-512. Cassano, A. E.; Alfano, 0. M. Photochemical Engineering Design Methods. A Rigorous Approach For Homogeneous Systems. In Photochemical Conversion and Storage of Solar Energy; Pellizzetti, E., Schiavello, M., Eds.; Kluwer: Dordrecht, 1991; pp 623-652. Cassano, A. E.; Brandi, R. J.;Alfano, 0.M. Photoreactor Modeling Fundamentals: A Rigorous Approach to Engineering Design. Trends Chem. Eng. 1994a,in press. Cassano, A. E.; Martin C. A.; Alfano, 0. M. The Radiative Transfer Equation Applied to Photochemical Reactors Revisited. Trends Photochem. Photobiol. 1994b,3,55-79. Cerda, J.;Irazoqui, H. A.; Cassano, A. E. Radiation Fields Inside an Elliptical Photoreflector with a Source of Finite Spatial Dimensions. AZChE J . 1973,19,963-968.

2200 Ind. Eng. Chem. Res., Vol. 34, No. 7, 1995 Cerda, J.; Marchetti, J. L.; Cassano, A. E. Radiation Efficiencies in Elliptical Photoreactors. Lat. Am. J. Heat Mass Transfer 1977,1,33-63. Chiltz, G.; Goldfinger, P.; Huybrechts, G.; Martens, G.; Verbeke, G. Atomic Chlorination of Simple Hydrocarbon Derivates in the Gas Phase. Chem. Rev. 1963,63,355-372. Claria, M. A.; Irazoqui, H. A.; Cassano, A. E. Modelling and Experimental Validation of the Radiation Field Inside an Elliptical Photoreactor. Chem. Eng. J . 1986,33,119-131. Claria, M. A.; Irazoqui, H. A.; Cassano, A. E. A Priori Design of a Photoreactor for the Chlorination of Ethane. AIChE J. 1988, 34,366-382. Clark, T. C.; Clyne, M. A. A. Kinetic Mechanisms in Nitrogen Chlorine Radical Systems. 2.Kinetics of Elementary Reactions of Nitrogen Trichloride and of the Dichloramino Free Radical. Trans. Faraday SOC.1970,66,372-385. Clyne, M. A.; Stedman, D. H. Recombination of Ground State Halogen Atoms. 11. Kinetics of the Overall Recombination of 1968,64,2698-2707. Chlorine Atoms. Trans. Faraday SOC. Daiton, F. S.;Lomax, D. A.; Weston, M. The Kinetics of The GasPhase Photochlorination of Trichloroethylene. Trans. Faraday SOC.1957,53,460-467. Davis, D. D.; Braun, W.; Bass, A. M. Reactions of C1 2P3,z.Absolute Rate Constants for Reaction with Hz, CHI, CZH6, CHZClz, Cpcl4, and c-CsHlz. Znt. J . Chem. Kinet. 1970,2,101-114. De Bernardez, E. R.; Cassano, A. E. Azimuthal Asymmetries in Tubular Photoreactors. Lat. Am. J . Heat Mass. Transfer 1982, 6,333-351. De Bernardez, E. R.; Cassano, A. E. A Priori Design of a Continuous Annular Photochemical Reactor. Experimental Validation for Simple Reactions. J . Photochem. 1985,30,285301. De Bernardez, E. R.; Cassano, A. E. Methodology for an Optimal Design of a Photoreactor. Application to Methane Chloro Derivatives Production. Ind. Eng. Chem. Process Des. Dev. 1986, 25,601-612. Doede, C. M.; Walker, C. A. Photochemical Engineering. Chem. Eng. 1955,62,159-178. Duderstadt, J.J.;Martin, R. Transport Theory; Wiley: New York, 1979. Esplugas, S.; Vicente, M.; Alfano, 0. M.; Cassano, A. E. Effect of the Reflector Shape on the Perfomance of Multi-Lamp Photoreactors Applied to Pollution Abatement. Znd. Eng. Chem. Res. 1990,29,1283-1289. Forbes, G.; Heidt, L. Optimum Composition of Uranyl Oxalate Solutions for Actinometry. J. Am. Chem. SOC. 1934,56,23632365. Gavalas, G. R. Long Chain Approximation in Free Radical Systems. Chem. Eng. Sci. 1966,21,133-141. Gosselain, P. A.; Adam, J.; Goldfinger, P. La Specificit6 des Halogenations Atomiques. 11. Les Regles Generales qui Determinent le Mecanisme et la Specificite des Halogenations Atomiques. Bull. SOC.Chim. Belg. 1956,65,533-548. Hatchard, C. G.; Parker, C. A. A New Sensitive Chemical Actinometer. 11. Potassium Ferrioxalate as a Standar Chemical Actinometer. Proc. R. Soc. London 1956,A235, 518-536. Heidt, L.; Tregay, G.; Middleton, F. Influence of pH Upon the Photolysis of the Uranyl Oxalate Actinometer System. J . Phys. Chem. 1970,74,1876-1882. Hutton, E.; Wright, M. Photoemissive and Recombination Reactions of Atomic Chlorine. Trans. Faraday Soc. 1965,61, 7889. Irazoqui, H. A.; Cerda, J.; Cassano, A. E. Radiation Profiles in an Empty Annular Photoreactor with a Source of Finite Spatial Dimensions. AIChE J . 1973,19,460-467. Irazoqui, H. A.; Cerda, J.; Cassano, A. E. The Radiation Field for the Point and Line Source Approximations and the ThreeDimensional Source Models. Applications to Photoreactions. Chem. Eng. J . 1976,11,27-37. Ivin, K. J.;Steacie, E. W. R. The Disproportionation and Combination of Ethyl Radicals. The Photolysis of Mercury Diethyl. Proc. R. SOC.London 1951,A208,25-42. Ivin, K.J.;Wijnen, M. H.; Steacie, E. W. R. Reactions of Ethyl Radicals. J . Phys. Chem. 1952,56,967. Kerr, J. A.;Trotman-Dickenson, A. F. The Reactions of Alkyl Radicals. 111. n-Butyl Radicals from The Photolysis of nValeraldehyde. J. Chem. SOC.1960,56, 1602-1608.

+

Knox, J. H.; Nelson, R. L. Competitive Chlorination Reactions in the Gas Phase: Hydrogen and C1-Cs Saturated Hydrocarbons. Trans. Faraday SOC. 1959,55,937-945. Kurtz, B. E. Homogeneous Kinetics of Methyl Chloride Chlorination. Znd. Eng. Chem. Process. Des. Dev. 1972,11, 332-338. Leemakers, J. A.; Dickinson, R. G. The Photochlorination of Tetrachloroethylene in Carbon Tetrachloride Solution. J.Am. Chem. SOC.1932,54,4648-4657. Leighton, W.; Forbes, G. Precision Actinometry with Uranyl Oxalate. J . Am. Chem. Soc. 1930,52,3139-3152. Martin, C. A.; Baltanas, M. A.; Cassano, A. E. Photocatalytic Reactors I. Optical Behavior of Titanium Oxide Particulate Suspensions. J . Photochem. Photobiol. A: Chem. 1993,76,199208. National Research Council. Potential Applications of Concentrated Solar Photons; National Academy Press: Washington, DC, 1991. Noyes, R. M. Wall Effects in Photochemically Induced Chain 1951,73,3039-3043. Reactions. J. Am. Chem. SOC. Noyes, W. A,; Leighton, P. A. The Photochemistry of Gases;Dover Publications Inc.: New York, 1941. Ollis, D. F.; Al-Ekabi, H. Photocatalytic Purification and Treatment of Water and Air; Elsevier Science Publishers: Amsterdam, 1993. Ollis, D. F.; Pelizzetti, E.; Serpone, N. Destruction of Water Contaminants. Environ. Sci. Technol. 1991,25,1523-1529. Otake, T.; Tone, S.; Higuchi, K.; Nakao, K. Light Intensity Profile in Gas-Liquid Dispersion. Applicability of Effective Absorption Coefficient. Kagaku Kogaku Ronbunshu. 1981,7,57-63. Ozisik, M. N. Radiative Transfer and Interactions with Conduction and Convection; Wiley: New York, 1973. Pai, S. Radiation Gas Dynamics; Springer Verlag: New York, 1966. Parker, C. A. A New Sensitive Chemical Actinometer. I. Some London 1953, Trials with Potassium Ferrioxalate. Proc. R. SOC. A220, 104-116. Pascuali, M.; Santarelli, F.; Porter, J. F.; Yue, P. L. Radiative Transfer in Photocatalytic Systems. Presented at Chisa '93, Praha, Czechoslovakia, 1993. Pellizetti, E., Serpone, N., Eds. Homogeneous and Heterogeneous Photocatalysis; Reidel: Dordrecht, 1986. Pellizetti, E., Schiavello, M., Eds. Photochemical Conversion and Storage of Solar Energy; Kluwer: Dordrecht, 1991. Poutsma, M. L.; Hinman, R. L. Chlorination Studies of Unsaturated Materials in Nonpolar Media. I. Solvent Effects on Radical Addition of Chlorine to Chloroethylenes. J . Am. Chem. Soc. 1964,86,3807-3814. Pritchard, H. 0.;Pyke, J. B.; Trotman-Dickenson, A. F. The Study of Chlorine Atom Reactions in the Gas Phase. J. Am. Chem. SOC.1955,77,2629-2633. Romero, R. L.; Alfano, 0. M.; Marchetti, J. L.; Cassano, A. E. Modelling and Parametric Sensitivity of an Annular Photoreactor with Complex Kinetics. Chem. Eng. Sei. 1983,38,15931605. Santarelli, F. One Dimensional Radiative Transfer in Planar Participating Media. Lat. Am. J . Heat Mass Transfer 1983,7, 35-49. Santarelli, F. Radiative Transfer in Photochemical Process. In Photoelectrochemistry,Photocatalysis and Photoreactors;Schiavello, M., Ed.; Reidel: Dordrecht, 1985;pp 549-559. Santarelli, F.; Stramigioli, C.; Spiga, G. Rigorous and Simplified Approach to the Radiative Transfer in an Absorbing and Anisotropically Scattering Slab with a Reflecting Boundary. Znt. J . Heat Mass Transfer 1980,23,853-860. Santarelli, F.; Stramigioli, C.; Spiga, G.; Ozisik, M. N. Effects of Scattering and Reflection of Radiation on Batch Photochemical Reaction in a Slab Geometry. Znt. J . Heat Mass Transfer 1982, 25,57-62. Schiavello, M., Ed. Photoelectrochemistry, Photocatalysis and Photoreactors; Reidel: Dordrecht, 1985. Schiavello, M., Ed. Photocatalysis and Environment; Kluwer: Dordrecht, 1988. Serpone, N.; Pellizetti, E. Photocatalysis: Fundamentals and Applications; Wiley: New York, 1989. Siegel, R.; Howell, J. R. Thermal Radiation Heat Transfer, 3rd ed.; Hemisphere: Washington, DC, 1992.

Ind. Eng. Chem. Res., Vol. 34,No. 7, 1995 2201 Slattery, J. C. Momentum, Energy, and Mass Transfer in Continua; McGraw-Hill: New York, 1978. Spadoni, G.; Bandini, E.; Santarelli, F. Scattering Effects in Photosensitized Reactions. Chem. Eng. Sci. 1978,33,517-524. Spadoni, G.; Stramigioli, C.; Santarelli, F. Influence of a Reflecting Boundary on an Heterogeneous Photosensitized Reactions Within a Plane Slab. Chem. Eng. Commun. 1980,4, 643-649. Spiga G.; Santarelli, F.; Stramigioli, C. Radiative Transfer in an Absorbing and Anisotropically Scattering Slab with a Reflecting Boundary. Znt. J. Heat Mass Transfer 1980,23, 841-853. Steacie, E. W. R. Atomic and Free Radical Reactions, 2nd ed.; Reinhold: New York, 1954; Vol. 1. Stramigioli, C.; Spadoni, G.; Santarelli, F. Photosensitized Reactions in Absorbing-Scattering Media Within a Plane Slab. Znt. J. Heat Mass Transfer 1978,21, 660-662. Stramigioli, C.; Santarelli, F.; Spiga, G.; Vestrucci, P. Sun Driven Photochemical Reaction in a Horizontal Plane Slab. Chem. Eng. Commun. 1982,15,205-214. Tymoschuk, A. R.; Alfano, 0. M.; Cassano, A. E. The Multitubular Photoreactor. 1. Radiation Field for Constant Absorption Reactors. Znd. Eng. Chem. Res. 1993a, 32, 1328-1341. Tymoschuk, A. R.; Alfano, 0. M.; Cassano, A. E. The Multitubular Photoreactor. 2. Reactor Modeling and Experimental Verification. Znd. Eng. Chem. Res. l993b, 32, 1342-1353.

van de Hulst, H. C. Light Scattering by Small Particles; Wiley: New York, 1957. Vicente, M.; Alfano, 0. M.; Esplugas, S.; Cassano, A. E. Design and Experimental Verification of a Tubular Multilamp Reactor for Water and Waste-Water Treatment. Znd. Eng. Chem. Res. 1990,29, 1278-1283. Vincenti, W. G.; Kruger, C. H., Jr. Introduction to Physical Gas Dynamics; Wiley: New York, 1965. Walling, C. Free Radicals in Solution; Wiley: New York, 1957. Whitaker, S. Fundamental Principles of Heat Transfer;Pergamon Press: New York, 1977. Yokota, T.; Iwano, T.; Deguchi, H.; Tadaki, T. Light Absorption Rate in a Bubble Column Photochemical Reactor. Kagaku Kogaku Ronbunshu 1981, 7, 157-163.

Received for review December 23, 1994 Accepted April 11, 1995 @

I39407621

Abstract published in Advance ACS Abstracts, June 1, 1995. @

Photoreactor Analysis and Design: Fundamentals ... - ACS Publications

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