500 Znd. Eng. Chem. Res. 1995,34, 500-509

Scattering Effects Produced by Inert Particles in Photochemical Reactors. 2. A Parametric Study

Maria I. CabreraJ Orlando M. Alfano: and Albert0 E. Cassano*9* INTEC,f UNL,“ and CONZCET,l Giiemes 3450, (3000) Santa Fe, Argentina

A model for heterogeneous (solid-fluid) photochemical reactors, previously developed and validated experimentally, has been used to investigate the effects produced by absorption and scattering on different variables associated with the reactor performance. With this purpose a parametric study was conducted analyzing the effects engendered by changes in the scattering inert center concentration, scattering center characteristics, and radiation absorbing species concentration on the following properties of the heterogeneous system: (i) spatial and directional distribution of radiation intensities, (ii) local volumetric rate of energy absorption, (iii) reactor volume averaged rate of reaction for a simple reaction occurring in the fluid phase, and (iv) radiation absorption efficiency inside the reactor. The effect of adopting different models of the scattering phase function for computing the radiative transfer equation was also studied. Operating conditions under which scattering effects may be significant were defined. It was found that there exist combinations of the optical parameters where addition of inert scattering centers may reduce the rate of reaction in as much as 50%.

I. Introduction

In a previous work (Alfano et al., 1995) we have developed a mathematical model for predicting the radiant field in a heterogeneous reacting system where radiation absorption and scattering occur. In that paper the theoretical aspects of the problem were presented and the numerical method of solution was described. The system parameters for solving the radiative transfer equation were obtained, and predicted results were validated with experiments. Bench scale verifications were carried out by studying the effects produced by addition of scattering centers (inert, transparent, spheri- cal, silica beads) to a liquid solution where, upon radiation absorption, the homogeneous photodecompo- sition of the uranyl oxalate complex was measured. The experimental device was an isothermal, well-stirred, continuous reactor operated inside a batch system with recycle and irradiated by a tubular lamp and a parabolic reflector.

To evaluate the radiation intensity of the photoreactor system, a planar geometry was adopted (Figure 1). The model was developed under the following assumptions (Alfano et al., 1995): (i) the radiation field is at steady state and there is no emission of radiation inside the reactor; (ii) scattering is independent; (iii) the absorption and scattering coefficients are uniform throughout and only depend on wavelength, and radiation absorption species concentration and particle concentration respec- tively; (iv) only one direction of radiation propagation is considered (x) and this propagation occurs with azimuthal symmetry (a single angular direction 8, with angular symmetry in 4); (v) the reactor surface of radiation entrance (at x = 0) receives polychromatic radiation in a diffuse manner (independent of 6); and (vi) the opposite plate of the reactor (at x = L) is transparent. Experiments were designed to fulfill these

~~ ~~

* To whom correspondence should be addressed. +Research Assistant from UNL. f Professors (UNL) and CONICET Research Staff Members. 8 Instituto de Desarrollo Tecnoldgico para la Industria

II Universidad Nacional del Litoral. Consejo Nacional de Investigaciones Cientificas y Tecnicas.

Quimica.

,

fL’0

Figure 1. Coordinate system for radiative transfer.

conditions. The derived equation was

and where p = cos 8. The local volumetric rate of energy absorption (LVREA)

that is needed to write the rate of reaction was obtained from the radiation intensities according to

1

ea,(x) = 2 m A j ~ , ( x , p ) 44 (4) /I=-1

Finally, the expression employed to compute the reactor volume averaged rate of oxalic acid decomposi- tion was

0888-588519512634-0500$09.00/0 0 1995 American Chemical Society

Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995 501

Scattering was modeled as anisotropic and indepen- dent, produced by spherical particles having a specular, partial reflection according to the following distribution function (phase function):

In this paper the scattering phenomenon inside the heterogeneous reactor is studied in detail, analyzing the effects produced on the system by changes in its more important parameters. Using the scattering inert cen- ter concentration and the radiation absorbing species concentration as independent variables, the resulting consequences on the following aspects will be de- scribed (i) directional (angular) and spatial distribution of radiation intensities inside the reactor, (ii) spatial distribution of the volumetric rate of energy absorption inside the reactor, (iii) ratio of the reactor volume averaged rates with and without scattering, and (iv) radiation energy absorption efficiency inside the reactor. Also, changes in the reactor performance predictions are discussed when different forms (models) of the scatter- ing phase function are used to solve the radiative transfer equation. Finally, we report results obtained when the scattering characteristics of the scattering inert centers are modified.

11. Angular Distribution of the Radiation Intensity

The directional and spatial distribution of the radia- tion intensity I(x,f3) for the heterogeneous system (with absorption and scattering) is obtained by solving eqs 1-3 and afterward performing an integration over all useful wavelengths:

The results are presented and analyzed reporting the angular variations of the radiation intensity a t fixed positions along the x coordinate, when the concentration of the scattering centers and the concentration of the radiation absorbing species are changed. The particles are silica spherical beads and their concentration C, was varied from 0 (homogeneous medium) to 6.0 x 10- g ~ m - ~ . Alfano et al. (1995) performed a study of the silica particles. A sample was observed in the micro- scope, and the following results were obtained: (i) minimum diameter = 37 pm, (ii) maximum diameter = 75 pm, (iii) mean diameter = 58 pm, and (iv) standard deviation of the size distribution = 13 pm. Considering the particle mean diameter and the particle volume fraction employed in the different experimental runs, the number of scattering centers per cubic centimeter was varied from 0 (no particles) to 3.2 x lo4 particles per cm3. The concentration of the uranyl oxalate complex C,, was changed from 1 x mol ~ m - ~ . Both values were chosen in agreement with operating conditions used in the quoted paper.

mol ~ m - ~ which produces rather high radiation absorption. Using polar coordinates, the variations with the angular position are shown for three different distances along the x coordinate: (i) at the reactor radiation entrance (x = 01, (ii) at the reactor middle position (x = 0.05 m), and (iii) at the reactor end (x = 0.10 m).

In Figure 2(a) one can observe results for a medium without particles (only absorption occurs) which are

B

to 5 x

Figure 2 shows results for C, = 5 x

a n n

2 I 0 1 2

I XIO' (einstein s-' cm-2 sr-I 1

2 I 0 1 2

I x 10' (einstein s-' cm-2 s i ' )

90"

180" 00 2 I 0 2

I x IO* (einstein s-' cm-2 sr-l) Figure 2. Angular distribution of radiation intensity for Cur= 5 x g ~ m - ~ , and (c) C,, = 6 x g ~ m - ~ . (-) x = 0; (- -) x = 5 cm, and (- - - ) x = 10 cm.

mol ~ m - ~ : (a) C,, = 0, (b) C,, = 3 x

included to be used as a comparison basis for discussing the consequences of incorporating scattering centers to the system. At x = 0 the plot reproduces the model boundary condition; Le., at the reactor entrance there is a uniform and diffuse radiation inlet and, conse- quently, the angular distribution indicates an intensity value that is constant. At the reactor center (x = 0.05 m) due to the presence of a participating medium (absorption) there is a reduction of the radiation inten- sity for all angles. We can note that for angles lying between 80" and 90" changes in intensities are more pronounced. In theory, for a planar, one-dimensional medium, at a constant value of x , when the angle between the x-axis and the analyzed direction (angle 6 ) is small, photons travel a shorter distance than when the angle approaches n12; an increase in this distance augments the optical thickness and the result is a higher absorption. In reality, these results correspond to an ideal situation: the radiation distribution model was developed, as indicated before, for a planar, one- dimensional medium; hence, as far as the angular directions are concerned, in some cases (for example, it is evident for 8 > ~ 1 4 ) we are computing radiation

502 Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995

absorption for optical path lengths that may have part of their trajectory outside the reactor boundaries (be- cause the actual reactor has a finite diameter). At the reactor end a similar situation can be observed, but in this case all the computed values of the radiation intensities are smaller.

In Figure 2(b) the computed results correspond to a case where some scattering is present (Cmp = 3 x lop3 g ~ m - ~ ) . Now there will be photons that, as usual, travel in the forward direction but, due to the existence of scattering, other photons will fly backward. At x = 0, once again, one observes for 0 5 8 5 zI2 and 3x12 I 8 I 237 (or p > 0) the model boundary condition (which is represented in the first quadrant making use of sym- metry considerations). At the same position the model indicates the presence of a radiation field for directions corresponding to d 2 5 8 I n and x I 8 5 3nl2 (or p < 0) which are represented in the second quadrant; this is the result of backward scattering produced by the inert particles; all these photons are escaping from the reactor and are lost for the purpose of producing reaction. For this reason, the observed reaction rate inside the reactor will be slower than in the case of pure absorption. At the middle position (x = 0.05 m) the angular diagram of radiation intensities is similar to the previous one but the intensity values are smaller; this reduction is due to two effects: (i) radiation absorption and (ii) radiation scattering. It should be noticed that, in this case, part of those photons that are traveling backward most probably will not escape from the reactor because they can be absorbed in the medium from x = 0.05 to x = 0.0. Finally, at the reactor outlet ( x = 0.10 m) an almost negligible radiation intensity angular distribution for values of p > 0 (see first quadrant in the plot) and a zero, uniform value for p < 0 (second quadrant) can be observed. This last result reflects the boundary condition (eq 3) used to describe the system at x = L [I(L,8) = 01. In fact it may be recalled that a t the second face of the planar reactor there is no radiation inlet (a non emitting boundary) and reflections were neglected (a transparent boundary).

Figure 2(c) presents the results of a similar analysis but for a system with more important scattering (Cmp = 6 x g cm-3). There can be observed important quantitative differences: a greater scattering produces (i) a greater reduction in the number of photons that are moving forward > 0, in the first quadrant) and (ii) a greater intensity in all the photons that are moving backward ( p < 0). Comparing Figure 2a-c, it may be seen that the last case represents a condition where the escape of photons from the front plate of the reactor is greater. Compared with the situation described by Figure 2a,b these results indicate that the volume averaged reaction rate of the homogeneous reaction should be further decreased. Experiments have con- firmed these findings (Alfano et al., 1995).

It may be important to remark that results concerning the distribution of radiation inside photoreactors have been usually presented employing a different property. It is very common to use the following variable:

G = J I ~ Q R

which is the Incident radiation. Radiation intensities have units of energy per unit area, unit time, and unit solid angle of propagation whereas the incident radia- tion (in spite that is frequently and improperly called intensity) has units of energy pes unit area and unit

time. The radiative transfer equation is always written in terms of the radiation intensity that is the primitive concept. From this variable one can obtain all the others such as the incident radiation, the radiation flux, and the rate of radiative energy absorption per unit volume (or the LVREA). When the primitive variable is used, the spatial distribution (x for the one-dimen- sional model) and the directional distribution (8 when the system has azimuthal symmetry) can be analyzed. If one uses variables where the directional (angular) dependence has been integrated, the only possible representation is reduced to the spatial dependence. The complete representation furnished by the radiation intensity permits understanding in greater detail the qualitative and quantitative aspects of the radiation field inside the reactor; otherwise, visualization may be obscured when other macroscopic variables are used. This aspect is even more critical here because, generally, macroscopically averaged quantities such as the average rate of reaction or the average rate of radiation absorp- tion are the observed properties in the experimental devices. These are needed properties for reactor design, but they are better understood when their analysis is enlightened by the previous knowledge of the radiation intensities.

111. Rate of Radiant Energy Absorption Profiles The spatial variation of the local volumetric rate of

energy absorption (LVREA) will be analyzed in this section. This variable can be readily obtained by integration over the solid angle of radiation propagation (4x1 of the existing radiation intensities inside the reaction system. In this case, as indicated in eq 4, due to the azimuthal symmetry, only one angular variable is required (either 8 or p). For polychromatic radiation, eq 4 must be integrated over the useful wavelengths:

This variable is the one that must be used to compute the reaction rate inside the photoreactor (Irazoqui et al., 1976; Bandini et al., 1977).

In Figure 3 we represent the LVREA as a function of the spatial coordinate x , for an absorbing species concentration (C,) equal to 5 x mol cm-3 and three different mass particle concentrations: C,, = 0, 3 x

High radiation absorption is the condition prevailing in the figure; it corresponds to the higher radiation absorbing species concentration investigated. In the homogeneous case (Cmp = 0) the variation of the LVREA with position is rather steep. At constant absorption, addition of particles to the system (centers of scattering which augment the scattering coefficient) produces an increase in the extinction coefficient (PA); i.e., the optical thickness increases and, consequently, the plot shows curves that become steeper. It must be noted that the absorbed energy is higher in the first middle part of the reactor and lower in the second. To some extent this may appear as an unexpected result: scattering in- creases the rate of energy absorption in one part of the reactor (close to x = 0) and decreases it in others (close to x = L). Precisely scattering is responsible for this behavior because photons that are scattered backward may still be absorbed by the medium. At the reactor radiation entrance the increase in the rate of energy absorption is as large as 12%. On the other hand, at

and 6 x lop3 g ~ m - ~ .

Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995 603

- - ‘UJ

E ?

V

c a 2 2.5

a

.- t

.- W

00

0 X

h

x

a, W 0

0 .o

c,, = 5 x mole cm-3

0 2 4 6 8 IO

x (cm) Figure 3. Distribution of local volumetric rate of radiant energy absorption for Cur = 5 x loW6 mol ~ m - ~ .

the opposite side (rear plate, x = L), scattering produces the maximum decrease in the LVREA. One can note that in the first 4 cm of the reactor length the value of ea is decreased by a factor of 10 and at the reactor end (x = L) the absorption of energy is almost negligible. In short, (i) the LVREA profiles are very steep and (ii) differences between both extremes of scattering (maxi- mum and zero) are not significant.

Similar results for a simpler system were reported by Stramigioli et al. (1978) and Spadoni et al. (1980). Strictly speaking these results are not comparable with ours; these authors report results for the LVREA by changing the scattering albedo (the ratio of the scat- tering coefficient and the extinction coefficient) main- taining the optical thickness constant; this implies that any change in the albedo (for instance by increasing the scattering coefficient) must be compensated by an equivalent decrease in the absorption coefficient. The results illustrated in Figure 3 show that when one increases the concentration of scattering centers (C,,) for a constant value of the system radiation absorption (C, = constant) the optical thickness of the system increases unavoidably.

IV. Volume Averaged Reaction Rates In this section we will analyze the effects produced

on the average homogeneous reaction rate by addition of inert particles; the study will be made for different radiation absorbing conditions.

Heterogeneous Reaction Rate. From now on, when scattering centers are added to the system, the homogeneous reaction rate will be denominated hetero- geneous reaction rate. Equation 5 provides the tool for the analysis. The local values of the polychromatic reaction rates are averaged over the reactor volume (from 0 to L in the one-dimensional model). These values can be plotted as a function of the radiation absorbing species concentration. Once more, the results are investigated for different particle concentrations (Cmp = 0, 3 x and 6 x low3 g ~ m - ~ ) . The homogeneous case (Cmp = 0) is taken as a reference.

Figure 4 depicts the results. The first important conclusion is that, in all cases, increasing the concentra-

/ I I I

1 2.5 5.0 7.5 1c

c,, x lo6 (mole 0

Figure 4. Volume-averaged reaction rate as a function of uranyl ion concentration for C,, = 0, 3 x

tion of the absorbing species always increases the average reaction rate regardless of the scattering condi- tions. At the beginning this increase is very important, but a t medium or higher concentrations the effect is leveled off reaching at the highest concentrations an almost asymptotic value. This behavior is characteristic of a well-stirred photochemical reactor: increasing the optical thickness (by increasing the absorption coef- ficient) increases the rate (Cassano and Alfano, 1989). It must be noted that this is not the case if perfectly stirring conditions are not fulfilled. When scattering is present, the pattern of the curves is very alike but the reaction rates are always slower than in the homogeneous case. When the concentration of inert scattering centers is increased, this effect becomes more significant. A similar result was described in an experimental study reported by Riva et al. (1981). These results should be interpreted considering those previously discussed in Figure 3. In spite of the observed changes in the curves of the LVREA, the volume averaged reaction rate is lower when scattering is present. An explanation can be found if one thinks in terms of the known effects produced by scattering: some of the photons will always escape from the system. Then, under equal conditions of absorption the average absorption rate of photons will be higher in the homo- geneous system. In other words, the difference between the incoming photons (at x = 0) and the outgoing photons (at x = L) will be maximum for the homoge- neous reactor because in the heterogeneous one some of them are lost due to scattering. The higher absorp- tion rate at the reactor radiation entrance observed in Figure 3 does not compensate the escape of photons at x = 0 and the lower absorption rates existing in other parts of the reactor.

This effect has a different magnitude depending on the concentration of the absorbing species. At low radiation absorbing species concentrations there is a very small difference because if the reactor optical density is very thin, the probability of absorption is low regardless of the existence (or not) of scattering. One must observe that these conditions (with a very small concentration of the absorbing species) are somewhat in conflict with the one-dimensional model. When absorption is low, under normal particle concentrations,

and 6 x g ~ m - ~ .

504 Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995

r ! \

I I I

0 0 2.5 50 75 100

c,, x 1o6(mole ~ m - ~ ) Figure 5. Ratio of reaction rate to homogeneous reaction rate as a function of uranyl ion concentration for C,, = 0, 3 x and 6 x g ~ m - ~ .

the extinction coefficient will not be very high. The one- dimensional model may provide an appropriate repre- sentation of a problem when the optical thickness is not too small; otherwise, the characteristic optical length becomes too large (and comparable with the other system dimensions) and a single spatial coordinate may not be sufficient to describe the scattering phenomenon appropriately.

When the concentration of the absorbing species is greater, it becomes important whether or not the photons may or may not be scattered out of the reactor. It will be seen that this phenomenon is reversed at very high concentrations of the absorbing species and the differences tend to diminish.

Ratio of Heterogeneous to Homogeneous Reac- tion Rates. One of the most interesting results is the analysis of the way in which the ratio of both reaction rates is changed with modifications in the concentration of the radiation absorbing species. Figure 5 shows these results that are useful to indicate, in a relative way, how important the effect of scattering is and how it changes with changes in the scattering center concentration. The figure provides a representation of ( R o x ) / ( R o x ) ~ ~ ~ vs C,. The different curves are made using the concentration of the scattering centers as a parameter. When C,, = 0 we have the ratio equal to 1 or the homogeneous reaction rate. In all cases the reported data are aver- aged over the reactor volume.

Clearly, the maximum value is that obtained with the homogeneous reaction. This is the upper limit for all perfectly mixed systems. At very low radiation absorb- ing species concentration the effect, as seen before, is almost negligible; i.e., the ratio approaches the maxi- mum value. Both reaction rates are very low and the ratio of both values is close to 1. When absorption is very low, the photon mean free path is so large com- pared with the reactor dimensions that photons are lost from the reactor space regardless of the effect produced by the scattering centers. When the concentration of the absorbing species is very high, the figure indicates that there is a tendency for the ratio to approach, in a very slow manner, the value of 1. In practice, for the system used as an example (photodecomposition of the

uranyl oxalate) the second value of 1 cannot be reached under practical conditions and, what is also important, many of the reaction parameters and physical properties would lose validity. In any event, clearly, as a theoreti- cal speculation, for a very high radiation absorption the limit of 1 should exist. In physical terms this means that the absorption is so high that even if a photon is scattered, it will be absorbed by the medium before it escapes from the reactor boundaries. Scattering mainly changes the place where the photon will be absorbed. Under these conditions, the photon mean free path is very short compared with the reactor dimensions. Obviously, this limiting condition for the very high concentrations will be approached in a different manner depending upon the particle concentration.

In the figure it is observed that there exists an intermediate concentration for which the ratio of both reaction rates reaches a minimum value. For the system investigated this value is reached for an absorb- ing species concentration between 4 x and 5 x lop6 mol cmP3. For the intermediate particle concentration the heterogeneous reaction rate is decreased to 93%, and for the higher particle concentration the value is de- creased to 86%. Clearly, in same cases, if scattering is not considered, the error may be significant. This effect will be even more important if particles of smaller size were used. If this is the case, one should expect that for the same particle mass concentration the scattering coefficient should be greatly increased.

V. Absorption Efficiencies in Heterogeneous Systems

In any case it should be clear by now that the main question at stake is how much energy is lost due to scattering. To answer this question we may resort to the idea of photoreactor efficiency that was developed in detail by Cerda et al. (1977). A n overall efficiency was defined as follows:

The energy yield or efficiency is defined as the ratio between the amount of desired product obtained in the reactor and the amount of energy required to operate the radiation source. It can be obtained as the product of the efficiencies corresponding to four different steps: (i) the lamp emission yield ( 7 ~ ) which is radiation energy emitted by the lamp divided by energy consumed by the lamp, (ii) the reactor radiation incidence yield (71) which is radiation energy arrived at the reactor boundaries divided by energy emitted by the lamp, (iii) the reaction space radiation absorption yield (VA) which is radiation energy absorbed in the reaction space divided by energy arrived at the reactor, and (iv) the reaction product yield (overall quantum yield, qp) which is reaction desired product formed divided by radiation energy absorbed in the reaction space.

In order to analyze the effect produced by scattering in the efficiency of a given process, one needs to consider a system where the following aspects are well-defined: (i) the operating characteristics and dimensions of the radiation source (to compute VE), (ii) the geometric arrangement and optical characteristics of the lamp- reflector (if used)-reactor (to compute V I ) , and (iii) the overall quantum yield (to compute ~ p ) . If this informa- tion is available, then these three yields will be known. Experimentally, one can measure 7 and obtain the absorption yield 7 ~ . Theoretically, 7~ can be also

Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995 606

30"

0"

330"

1 50°

180"

210°

Figure 6. Angular distribution of phase function for particles with specular partial reflection (-), isotropic phase function (- - -), and phase function for particles with diffuse reflection (- -).

predicted as it will be shown below. It may be noted that for a given lamp-reactor-reaction system 73, 71, and 7p are fixed and that according to the radiation absorption and scattering conditions the value of VA will change. Precisely, the absorption efficiency gives good information about the energy that can be effectively used in the reactor. Comparing the absorption efficien- cies of cases where there is scattering and cases where there is not, one can reach conclusions about scattering effects on the reactor performance. The prediction of the energy absorption efficiency can be obtained as the ratio of the absorbed energy in the reactor volume (per unit time) and the arriving energy at the reactor boundary of radiation entrance (per unit time). In mathematical terms,

(11)

Effect of the Concentration of the Radiation Absorbing Species. The energy absorption efficiency will be studied using the concentration of the scattering centers as a parameter and changing the concentration of the radiation absorbing species. As before the parameter C,, will be changed from 0 to 3 x and 6 x g ~ m - ~ . One may expect that these results should be similar to those represented in Figure 4. Actually, it can be shown (see the Appendix) that if the overall quantum yield were independent of wavelength, VA = y(Rox)with y being a proportionality constant.

Radiation absorption efficiencies were computed as a function of the radiation absorbing species concentra- tion. A plot very similar to Figure 4 was obtained. These results confirm that, for this particular reacting system, the linear relationship between l ; l ~ and (RoJ is a very good approximation. The proportionality con- stant was obtained in the Appendix. The results indicate that (i) under low concentrations of the radia- tion absorbing species the energy absorption efficiency is always low regardless of the concentration of scat- tering centers, (ii) increasing the concentration of the radiation absorbing species increases the efficiency up

to a point where an almost asymptotic value is reached, (iii) this asymptotic value is lower for the largest concentration of scattering centers, and (iv) under practical concentrations of the absorbing species the energy absorption yield reaches a value equal to 1 only for the homogeneous case.

Effect of the Phase Function. Phase functions are very difficult to measure. In general, different ap- proximations are used which correspond to models of the real phenomenon. Up to now we have been using a phase function that is the theoretical result of consider- ing that our scattering centers are (i) spherical particles (which is true), (ii) transparent to radiation in the wavelength interval of interest (which is also closely satisfied), (iii) of a size that is large compared with the used radiation wavelength (which is true), and (iv) producing a specular, partial reflection (which is a limiting case and henceforth an approximation). We will denominate this condition case 1. It seems conve- nient to investigate how much these predicted results would change if other types of phase functions should have been chosen. Let us look at the following possible cases: case 2 (iv'), scattering centers producing an isotropic scattering; and case 3 (iv"), scattering centers producing a diffuse type of reflection. Figure 6 gives a graphic representation, in polar coordinates, of the phase functions corresponding to the three cases (Siege1 and Howell, 1992).

The phase function corresponding to case 1 presents a very high angular anisotropy (note that a factor of 0.1 has been used in order to plot it in Figure 6). As can be observed, the phase function has a very sharp peak pointed forward; i.e., the function takes on very large values ( d o ) for small angles and decreases its value to almost 0 for large angles. This type of behavior facilitates the transport of photons in the positive x direction. In case 2 none of the directions are privileged because scattering is angularly isotropic; it models some sort of neutral situation. In case 3 again, a highly anisotropic scattering is represented but in this situa- tion scattering produces a motion of the photons toward the back (front of the reactor); the phase function has a value of 0 for 80 = 0 and reaches its maximum value of 2.67 for 80 = n. Cases 1 and 3 model limiting conditions

506 Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995

IO

0 5

0 0 0 0 50 IO 0

c,, to6 (mole cm-3 Figure 7. Absorption efficiency as a function of uranyl ion concentration for C,, = 0 and 6 x g ~ m - ~ . (- --) Phase function for particles with specular partial reflection; (- * -) isotropic phase function; (- - -) phase function for particles with diffuse reflection.

and only under very idealized situations the reflection will be purely specular or purely diffuse; one should expect that in real cases both models be an approxima- tion.

Equation 6 provided the mathematical representation of the phase function for case 1. Those corresponding

o case 2 and case 3, according to Siege1 and Howell 1992), are

Pcuo) = 1 (12)

The angle po can be expressed as a function of the wo angular variables 8 and 4. However, our model has

been written in terms of only one angular variable (0) and, consequently, the 4 dependence cannot be incor- porated. As indicated by Ozisik (1973) knowing p b o ) , the phase function for a system with azimuthal sym- metry can be obtained from

N

xanPnf&) Pn@’) with a, = 1 (14)

where the coefficients a, are obtained by expansion of the phase function p k o ) in terms of the Legendre polynomials of order n [Pn@o)l.

It was found that the phase function calculation required a rather large number of terms. A computer program was developed that was successively adding more terms to the expansion until an acceptable differ- ence between the original and the expanded function was obtained. Since in case 1 the phase function predicts a very important forward scattering, which has a very sharp peak, satisfactory results required the use of a minimum of 30 terms. For case 2 we used eq 14 with N = 0 (see eqs 12 and 14). For case 3 the phase function was satisfactorily represented with N = 2.

n=O

Figure 7 shows the results for the three phase functions in a representation of the absorption efficiency vs the uranyl oxalate concentration for a situation where scattering is rather high (but not too high). The case of zero concentration of scattering particles is plotted as a reference. It can be observed that results are different for the three cases. It is interesting that case 1 shows the closer response to the homogeneous case. With respect to the isotropic scattering results, case 1 shows differences as high as 31%. Case 3 is the one that differs more significantly with respect to the homogeneous system and its differences with respect to the isotropic case are not very important; the maxi- mum observed difference is 8%. These results have a simple physical interpretation. The shape of the phase function for case 1 indicates a privileged scattering in the forward direction (almost as if it were a Dirac delta function). The resulting motion of photons is very similar to the case in which there is no scattering (those photons that are not absorbed move forward); moreover, with this type of scattering, having an almost single preferential direction, the escape of photons is greatly reduced. Case 3 represents a case of significant back- ward scattering that produces an important loss of photons in regions close to the radiation entrance; this phenomenon is responsible for the observed largest differences with respect to the homogeneous case.

Effect of the Scattering Coefficient. Up to now we have been reporting data showing the effect of adding scattering centers to a homogeneous reacting system. Different dependent variables were analyzed and the independent variable was always the concen- tration of the radiation absorbing species. The concen- tration of the scattering particles was used as a para- meter. The scattering centers were always the same: spherical, transparent particles of silica particles. As described by Alfano et al. (19951, these particles were Porasil A-B having a nominal diameter that range from 37 to 75 pm.

The used Porasil particles have a comparatively large diameter which result in a rather small scattering coefficient. In fact, in order to have a rather important scattering, we have raised the concentration of scatter- ing particles up to 6 x g ~ m - ~ . It is not difficult to foresee reacting systems where the size and character- istics of the scattering centers are very different. For example, the most widely used photocatalyst is titanium dioxide and, very often, particles of nominal size equal to 0.030 pm have been used; these particles are not transparent and most probably a very different type of phase function will have to be adopted. What is surely important is that these very small particles will produce a very high scattering even at rather low particle concentrations.

Consequently, in our computer simulations the prob- lem was investigated using particles of very significant scattering cross section. We adopted, again, transpar- ent (in the wavelength of interest) and spherical par- ticles. They are latex particles made of polyvinyltolu- ene; they have an average diameter of 2 pm.

Daniel and Incropera (1977) have measured the scattering cross section and the phase function of an aqueous suspension of polyvinyltoluene with the above characteristics. From their results we have obtained the value of the scattering coefficient (@A) as a function of wavelength in the range of 295-435 nm, and the results are indicated in Table 1. Actually, some ex- trapolation for values of J. below 370 nm was necessary because Daniel and Incropera data were taken for ;1 2

Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995 507

30

2 0 -

- 2 0 ' I I

0 T / 2 ll

8 0

Figure 8. Phase function for poly(vinylto1uene) spheres (after Daniel and Incropera, 1977). (bold -1 Geometric optics; (-1 Mie theory; (0) experimental data.

Table 1. Spectral Data for Suspensions of Porasil A-B (a) and Latex (b) Particles

295 276.0 31700 365 261.5 21800 305 273.5 30500 375 261.0 20500 315 270.0 28800 390 259.5 18900 325 268.5 27200 405 255.5 17000 335 266.0 25600 420 255.5 15700 345 264.0 24000 435 253.0 14700 355 262.5 23100

370 nm. In this case the extrapolation is not an important difficulty because (i) the main objective is to analyze the effect of scattering when the systems incorporate scattering centers of a large scattering coefficient and (ii) the extrapolation did not produce significant changes in the value of the coefficient. What is important to recognize is that when the obtained values for the scattering coefficient are compared with those measured for the silica particles (Alfano et al., 19951, one finds a value that is from 58 to 115 times greater depending upon the wavelength (see Table 1).

Next, we need the phase function. Daniel and In- cropera (1977) measured the phase function for the aqueous suspension of latex particles and found that their results were in good agreement with predictions obtained from the Mie theory (van de Hulst, 1957). Figure 8 represents these results as a plot of the logarithm of the phase function vs the angle 80. In this figure we have plotted (i) the experimental results of Daniel and Incropera (19771, (ii) the theoretical results obtained by application of the Mie theory, and (iii) the results derived from the geometric optics approach. Agreement is good. However, there are some differences between the Mie theory and the other results for two different regions of 80: (1) for very low values of 80 and (2) for some values of the 80 angle beyond d 4 . The first discrepancy is the consequence of the diffraction phe- nomenon that is not accounted for by the geometric optics model. The second region of differences appears in places where the results from the Mie theory show their typical oscillations, particularly when the phase function takes on very low values (they are from 1 to 2 orders of magnitude smaller than the previous ones). Results from the geometric optics approach have good agreement with these experimental values.

In this section, results from the geometric optics will be used to represent the phase function corresponding to the latex particles. Daniel and Incropera's experi- mental data were obtained for a wavelength equal to

I .o

6 0.5

0.0 0.0 2.5 5.0 7.5 10.0

c,,x106 (mole cm-3 Figure 9. Absorption efficiency as a function of uranyl ion cnncentratinn for the latex Darticle-uranvl oxalate svstem.

469 nm, and we are interested in a range defined by 295 I 13.5 435 nm. Then, a computer program was used that solves the Mie equations (Bohren and Huffman, 1983); one of the results is the form of the phase function; it was used to show that variations in the phase function are not very important in moving from A = 469 nm to A = 365 nm, a value that lies in the middle of our wavelength range of interest. As was discussed before, our one-dimensional model cannot use values of the phase function as a function of po directly because it was developed using a single angular vari- able. Once more, in order to transform the available information into one suitable for a model with azimuthal symmetry, an integration over the q5 coordinate must be carried out. The result can be expressed in terms of the Legendre polynomials and computations were per- formed as indicated before for eq 14. Due to the shape of the phase function, in this case, a minimum of 30 terms was required.

Results for the case of the latex particles are shown in Figure 9. The absorption efficiency is represented as a function of the radiation absorbing species concen- tration, using the particle mass concentration as a parameter. Comparing these results with those ob- tained for silica particles, one may notice that (i) the curves indicate a phenomenon of similar nature and (ii) there is a very important change in the magnitude of the observed effects. Thus, with a maximum latex particle mass concentration of 0.5 x g cm-3 (a value 6 times smaller than the minimum one used with silica particles) the absorption efficiency reaches a value as low as 46% of the equivalent homogeneous case (for C , = 2.0 x mol ~ m - ~ ) and 56% (for Cur = 10.0 x mol ~ m - ~ ) . The scattering effects are less important when (i) the particle mass concentration is very low, regardless of the radiation absorbing species concentra- tion, and (ii) the radiation absorbing species concentra- tion is very low regardless of the particle mass concen- tration.

Finally, these effects can be seen from a different angle if the results are analyzed in terms of the relative values of the volume averaged reaction rates. In Figure 10 a plot Of (Rox)/(Rox)HOM vs C , is presented. One can observe that when the Cur is very high, absorption within the system tends to decrease the magnitude of

508 Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995

C,,=0.5 x 10- 4q ~ 1 1 1 . ~

cmp= I . O X ~ o - ~ g c m - ~

0.25 0 .o 2.5 5.0 7.5 10.0

c,, x lo6 (mole cm-3 Figure 10. Ratio of reaction rate to homogeneous reaction rate as a function of uranyl ion concentration for the latex particle- uranyl oxalate system.

the scattering effects; however, the approximation of the curves to a ratio value of 1 is very slow and, evidently, it could not be reached for practical values of the homogeneous solution concentration. In other words, under practical conditions of radiation absorption, scat- tering effects by small particles may be important and cannot be neglected. On the other hand, for very low values of the radiation absorbing species concentration the ratio value of 1 is readily obtained but it decreases very rapidly when the C , increases. In the middle of both limits, a minimum value exists; this minimum value moves toward the lower concentrations of C , when the concentration of the scattering centers is increased. Summarizing, all curves present (i) a rapidly decreasing arm at the beginning (for C , going from low to medium), (ii) a minimum, which shifts towards lower radiation absorbing species concentrations when the concentration of the inert particles is increased, and (iii) a slowly increasing arm (for Cur going from medium to high). It may be important to remark once more that for not too high values of particle concentrations and practical values of radiation absorption, the decrease in the reaction rate may be as large as 46% of the corresponding value for the homogeneous case.

VI. Conclusions An extensive study of scattering effects in heteroge-

neous photoreactors has been presented. The reactor performance is analyzed investigating the effects pro- duced in a homogeneous reaction by addition of inert particles that act as centers of scattering. With this objective, a flat plate (parallel faces) photoreactor had been modeled. Using this model that had been previ- ously verified experimentally, scattering effects on dif- ferent dependent variables such as the radiation inten- sity, the local volumetric rate of energy absorption, the volume averaged reaction rate inside the reactor, and the system absorption efficiency were studied. The main conclusions are as follows:

1. Addition of scattering centers to the reacting medium produces scattering which results in a loss of radiation. The escape of photons is important a t the surface of radiation entrance due to the significant

contribution of backscattering. The resulting effect is a diminution in the average reaction rate inside the reactor. These effects can be clearly visualized when the angular distribution of radiation intensities is represented in polar coordinates for different reactor cross sections.

2. Addition of scattering centers increases the optical thickness of the reactor and produces steeper profiles of the local volumetric rate of energy absorption when it is represented as a function of the spatial variable. 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.

3. In a well-mixed photoreactor increasing scattering always decreases the reaction rate. This effect is small at low radiation absorption species concentration and moderate when such concentration is high. There exists an ample range of intermediate concentrations where one observes that the effect takes on its maximum value.

4. The concept of absorption efficiency was used to study these effects 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 well-mixed hetero- geneous reactor is always smaller than that of the equivalent homogeneous reactor (without scattering centers).

5. The absorption efficiency was also used to study the effects produced on the predictions by the adoption of different models for the phase function (distribution function for scattering). Results for the cases in which isotropic and diffuse reflection phase function models were used do not show important differences between them (in this paper never greater than 8%). Conversely, the case of a specular, partial reflection model differs in a significant way with the other two models, the differences being as large as 31%; at the same time, its results show the smallest differences when they are compared with the purely homogeneous case. These observations indicate that when the system is described by a nonisotropic phase function, its correct determina- tion is very important in order to obtain accurate predictions.

6. It was shown that there exist systems of practical interest where a high value of the scattering coefficient should be expected. In these cases the diminution in the value of the volume averaged reaction rate for the heterogeneous reactor, when compared with the homo- geneous equivalent system, may be important. For an ample range of concentrations of the radiation absorbing species, and medium to low mass concentrations of scattering particles, the decrease in the average reaction rate may be as large as 50%. In these cases, modeling of the radiation scattering in order to solve the radiative transfer equation will be unavoidable.

Acknowledgment

The authors are grateful to Consejo Nacional de Investigaciones Cientificas y TBcnicas (CONICET) and to Universidad Nacional del Litoral (UNL) for their support to produce this work.

Nomenclature a, = expansion coefficient, dimensionless C = molar concentration, mol mv3 C, = mass concentration, g m-3 d = diameter, m ea = local volumetric rate of radiant energy absorption,

G = incident radiation, einstein m-2 s-l I = radiation intensity, einstein m-2 sr-1 s-' L = length, m n = refractive index, dimensionless p = phase function, dimensionless P, = Legendre polynomial of order n q = radiative flux, einstein m-2 s-' R = reaction rate, mol m-3 x = axial coordinate, m Greek Letters p = volumetric extinction coefficient, m-I y = constant defined in eq A.5, m3 s mol-' 7;1 = energy efficiency 8 = spherical coordinate, rad K = volumetric absorption coefficient, m-l 2. = wavelength, nm p = the quantity cos 8, dimensionless 4 = hemispherical reflectivity, dimensionless 4' = directional specular reflectivity, dimensionless o = volumetric scattering coefficient, m-I dc = mass scattering coefficient, m2 g-l 4 = spherical coordinate, rad

Subscripts A = absorption E = emission HOM = relative to homogeneous phase I = incidence L = reactor length ox = relative to oxalic acid p = particle property P = production ur = relative to the uranyl ion A = a dependence on wavelength 0 = relative to the surface of radiation entrance

Special Symbols ( ) = average value

einstein mW3 s-l

= quantum yield, mol einstein-l

Ind. Eng. Chem. Res., Vol. 34, No. 2, 1995 609

Combining eqs A.2 and A.3 the following relationship is obtained:

Appendix In this appendix we derive the existing relationship

between the absorption efficiency and the reactor- volume-averaged reaction rate.

The reactor-volume-averaged rate of reaction is given by

(A.1)

Let us introduce one approximation. Let us assume that the spectral overall reaction quantum yield (which is a function of wavelength) can be substituted by an average value (independent of wavelength). Then

(A.2)

From the absorption efficiency (eq 11) one can write

The values of L and IOJ can be obtained for our particular case from the work of Alfano et al. (1995). From the same reference the averaged quantum yield is (@)A = 0.54. Substituting these values in eq A.4 one gets for the proportionality constant:

y = 3.96 x 10' m3 s mol-' (A.5)

Using this value of y , VA vs C, can be obtained from those represented in Figure 4 (Rod vs Cur with a good degree of approximation.

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Received for review March 16, 1994 Revised manuscript received September 23, 1994

Accepted October 18, 1994*

IE9401622

* Abstract published in Advance ACS Abstracts, January 1, 1995.