Simplified Modeling of Radiant Fields in Heterogeneous

Simplified Modeling of Radiant Fields in Heterogeneous Photoreactors. 1. ... and, what is more important, confidence limits for the final solution are...
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Ind. Eng. Chem. Res. 1997, 36, 4740-4747

KINETICS, CATALYSIS, AND REACTION ENGINEERING Simplified Modeling of Radiant Fields in Heterogeneous Photoreactors. 1. Case of Zero Reflectance Alberto Brucato* and Lucio Rizzuti Dipartimento di Ingegneria Chimica dei Processi e dei Materiali (DICPM), Universita` degli Studi di Palermo, Viale delle Scienze, 90128 Palermo, Italy

A strongly simplified model is proposed for the estimation of the radiant flow field, and related quantities such as the local volumetric rate of energy absorption, in heterogeneous photoreactors. The model is based on the assumption that a photon carrying energy greater than that of the catalyst band gap, when interacting with a catalyst particle, is invariably absorbed. The model equations allow, for simple geometries, analytical integral solutions to be obtained. These have the advantage of giving an immediate grasp on how the main physical parameters affect the radiation field: an important feature especially for engineering purposes. The case developed here is that of a plane slab, for both monosized catalyst particles and particles with size distributions. Notably, as no adjustable parameters are used, fully predictive results can be obtained. The model predictions are successfully compared with the results of original experimental tests, therefore leading to model validation. In Part 2 of this paper, another model is developed in which the assumption of zero reflectance of the particles is removed, yet allowing analytical solutions to be obtained. One of the key features of the two models is that their predictions bound the real behavior of irradiated photoreactors, as shown in the papers. Therefore, although only approximate solutions are obtained from each of the two models, the real behavior can be better approximated by some sort of interpolation between the two solutions and, what is more important, confidence limits for the final solution are obtained. 1. Introduction Modeling of photochemical reactors is quite a complex task: in fact to the usual difficulties met in the modeling of conventional chemical reactors, other complications also arise from the need to contemporaneously model the radiation flow field, which strongly affects reaction rates. In general the radiation flow field may in turn be influenced by the reactants’ concentration field; this is for example the case of homogeneous photoreactions where the local optical properties of the medium are strongly affected by reagents and products concentrations, which in turn depend on the radiation flow field as well as on fluid flow patterns (Alfano et al., 1986). In the case of heterogeneous photoreactors the absorbing medium consists of solid particles that absorb and scatter light independent of the presence of reactants that very often are transparent to radiation: this means that the radiation field is not coupled with the reactants concentration field and can then be computed in advance, resulting in a significant simplification of the whole modeling task (Alfano et al., 1986; Rizzuti and Brucato, 1988, 1991). On the other hand, the radiation field depends on the solids’ concentration field, which has to be either experimentally assessed or simulated by a suitable model. This simulation may be troublesome, as the catalyst concentration distribution depends in a complex way on the characteristics of the turbulent flow field inside photoreactors, but in any case it can be frequently afforded beforehand, thanks to the decoupling with the radiation field. * To whom correspondence should be addressed. E-mail address: [email protected]. S0888-5885(96)00259-X CCC: $14.00

It is also important to note that even when a uniform particle concentration exists, the modeling of the radiant field is still a very complex task, because of the complexity of the interaction between the radiation source and dispersed scattering-absorbing particles. In order to accomplish this task, a solution of the radiation transfer equation (RTE) is required, but due to the complexities of this equation, only numerical solutions are allowed for real cases. Moreover, the numerical solutions themselves are considerably complicated by the integro-differential nature of the RTE (Cassano et al., 1995; Rizzuti and Brucato, 1991; Santarelli, 1985). Other methods of solution, mainly based on Monte Carlo approaches, have been devised, but lengthy numerical computations are still required in order to obtain each particular solution (Pasquali et al., 1996). It is clear that, in order to obtain precise predictions of photoreactor performance, a “complete” model would be required, i.e., a model able to simulate solid particles concentration distribution, the related distribution of radiant energy and its local absorption rate, and perhaps the turbulent dispersion of both reactants and reaction products. This information would have to be coupled with accurate information on all kinetics involved in order to finally get accurate predictions of photoreactor performance. This is clearly to be regarded at present as an overwhelming task. On the other hand, applications of photocatalytic processes are getting fast to the industrial application stage so that the need arises for engineers to have soon at hand reliable tools for the design and development of industrial photocatalytic reactors. While the development of improved simulation tech© 1997 American Chemical Society

Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4741

niques for each of the above-mentioned physical phenomena goes on, it may thus be interesting to develop strongly approximate radiation field models, which by taking into account only the main factors affecting light distribution in photoreactors, give rise to simple, easyto-use, analytical solutions. Indeed, from an engineering point of view, the availability of simple, although approximate, equations is often what is really needed for the design and development of apparatuses. It is in fact such equations that give an immediate grasp and physical understanding of the role played by the key parameters on the whole process. This paper is actually devoted to the development of a simplified radiation field model, for the case of simple geometries and parallel irradiation of the particle suspension. In this model, the scattering of photons by particles is neglected. A second model is presented in the second part of the paper, where scattering is taken into account. 2. Radiant Field Modeling The model here developed is based on the following assumptions: (1) The catalyst particle size is significantly greater than photon wavelength; (2) diffraction phenomena are negligible; (3) a photon carrying enough energy to be useful for the promotion of photoreactions, when interacting with a catalyst particle, has a 100% probability of being absorbed (or a zero probability of being scattered). In view of this hypothesis, in the following this model will be referred to as the “zero reflectance model” (ZRM). The first assumption is based on the consideration that if particle size was comparable with the wavelength of photons of suitable energy in order to promote the photoactivity of the catalyst, i.e., smaller than 0.4 µm, it would be very difficult, if not impossible, to separate the catalyst particles from the reactor effluent. A full scale working photoreactor must then operate with much bigger particles if the catalyst is to be used as such rather than as a reagent. As a result of this assumption, the radiant field modeling can be performed by neglecting diffraction and related scattering phenomena, i.e., on the basis of simple geometric optics considerations. It is worth noting that this last simplification can be made for particle sizes greater than 5 λm/π (Siegel and Howell, 1992) where λm is the radiation wavelength in the particle material. As the wavelengths of interest for most photocatalysts are those smaller than λmax ) 400 nm (in the vacuum), confusing λmax with λm leads to the conclusion that the model should hold for particle sizes greater than about Dp,min ) 0.6-0.7 µm. Moreover, the refraction index of TiO2 is certainly greater than 1 so that neglecting this factor leads to a conservative value for Dp,min. The size range identified actually covers many practical applications with TiO2 particles, even when finely dispersed commercial TiO2 is used, because it is well-known (Martin et al., 1993) that, although the size of single crystals may be much smaller (even in the range of tenths of nanometers), the latter, once dispersed in water solutions, form aggregates which do often satisfy the above-stated condition. The second assumption can be regarded as a consequence of the first, provided that sufficiently diluted suspensions are dealt with, in order to ensure that statistical gaps between particles are also significantly greater than photon wavelength. It is worth noting that this is actually the case of slurry heterogeneous photo-

Figure 1. Schematic rapresentation of suspended particles for the ZRM in the case of plane geometry and parallel irradiation.

reactors which are usually operated with relatively diluted catalyst suspension. In regard to the third assumption, it is based on the consideration that the radiation field of interest is not the total radiation field, but rather the field of radiation characterized by a wavelength suitable for the promotion of the photoreaction (i.e., having an energy greater than the band gap of the semiconductor catalyst). This kind of radiation is usually effectively absorbed by the photocatalyst, as can be observed in published diffusereflectance spectra of photocatalysts, especially when the energy associated with the photon is significantly greater than the band gap of the catalyst itself. At energies only slightly greater than the band gap, this assumption is not completely verified but as it will be shown in Part 2 of this paper, the resulting lack of precision of model predictions may be considered acceptable, for engineering purposes, in many cases. Moreover, this is precisely one of the conditions to ensure that, under these conditions, the model represents one of the bounds of the real situation. Case of Plane Geometry with Parallel Irradiation. Let us consider the case of a plane slab irradiated with orthogonal, parallel radiation. With reference to Figure 1, if a unit cross section is considered and Ap is defined as the sum of all projected areas of particles statistically present in a volume of thickness ∆x, the following relationships apply to shades on planes 1 and 2:

Areas on plane 1 Ap 1 - Ap ) + 1 total shielded nonshielded Projected areas of particles in volume (1-2) Ap ) ApAp (1 - Ap)Ap + total already shielded newly shielded

Shielded areas on plane 2 As2 ) total

(1 - Ap)Ap + Ap shielded by shielded by particles in (1-2) particles in (0-1)

4742 Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997

(

Average radiant fluxes Gi at the ith plane

G ) G0 exp -

G0 G1 ) G0(1 - Ap) G2 ) G0(1 - Ap)(1 - Ap) and by extrapolation G3 ) G0(1 - Ap)(1 - Ap)(1 - Ap) etc. In general, one obtains:

Gx+∆x ) Gx(1 - Ap)

(1)

By indicating with ap the projected area of one particle and with np the number of particles per unit volume, it can be written, for the unit cross area, that

Ap ) np∆xap

(2)

Substituting in eq 1 and letting ∆x become arbitrarily small, one obtains:

dG ) -Gnpap dx

(3)

It is worth stressing that the way in which eq 3 was obtained is not entirely satisfactory. In particular the weak point is, with reference to Figure 1, that it was assumed that each particle in volume 0-1 was partially shading those in volume 1-2, as well as those in the subsequent volumes, while no account was made for its shading on the other particles in volume 0-1. It might be said that the volume thickness ∆x is arbitrary and thus it can be made small enough in order to make this last effect negligible, but this is still not entirely satisfactory. Let us rather state that it is evident that this problem clearly becomes more and more negligible while the slurry under consideration becomes more and more diluted, so that eq 3 can be accepted, provided that the solids concentration is small enough. Actually the authors are devising a probabilistic approach that allows the estimation of the deviations of eq 3 originated by the above-specified problem. This approach shows that these deviations are of the same order of magnitude as the percent volume concentration of the solid phase. As the photocatalyst concentrations actually employed in heterogeneous photoreactors are well below 1%, eq 3 can be retained here with negligible effects on the precision of results. Equation 3 can be easily integrated with the boundary condition G ) G0 for x ) 0, yielding

G ) G0 exp(-npapx)

(4)

By defining the volume shape factor R by vp ) RDp3, where vp is the volume of one particle, and the projected area shape factor γ by ap ) γDp2, and indicating with “m” the mass concentration of particles, it can be written:

np )

m ; FpRDp3

npap )

mβDp2 ) 3 FpRDp

Substitution into eq 4 finally gives:

m FpR/γDp

(5)

mx FpDpR/γ

)

(6)

Equation 6 shows that as a consequence of model assumptions, the predicted radiation field exhibits an exponential decrease while going deeper into the particle bed. In other words it follows a “Lambert-Beer type” law with an extinction coefficient proportional to the mass concentration of particles and inversely proportional to particle diameter. It is interesting to note that an integral solution very similar to eq 6 had already been developed (Rose and Lloyd, 1946), although with the aim of proposing an experimental technique for the estimation of particles size by means of light attenuation measurements. It is noteworthy that eq 6 holds true for a semi-infinite slab as well as for slabs of finite thickness L. In this last case it has a physical meaning only over the slab thickness and the last value in this region, i.e., the value at x ) L, represents the radiant energy lost through the rear wall of the photoreactor. It is clear that, the smaller this value, the greater the amount of useful radiation absorbed by the catalyst. Therefore, it is not surprising that several authors (Augugliaro et al., 1988, 1993; Pruden and Ollis, 1980) have found that the amount of reaction obtained in a given photoreactor increases with increasing catalyst concentrations, in the region of low catalyst concentrations, while at a certain value it reaches a plateau, giving rise to the concept of “critical” catalyst concentration. After that the achieved conversion can even show a slight decrease at even higher solids concentrations (Augugliaro et al., 1988). A similar behavior was also observed for gas-solid systems (Daroux et al., 1985). In light of eq 6, this behavior can be interpreted as follows: at low catalyst concentrations, substantial amounts of radiant energy are lost trough the rear wall of the reactor. As these decrease with increasing catalyst concentration, higher reaction rates are observed as a greater amount of radiation is utilized. When the concentration becomes high enough to practically extinguish the radiation before reaching the rear wall, all the radiation is being utilized and a further increase of catalyst concentration does not result in any increase of observed photoconversions. At higher catalyst concentrations, the layer in which light is extinguished becomes smaller and smaller and the masstransfer phenomena of reagents between the “dark” layers and the active irradiated layers begin to limit the reactants concentration in these last, giving rise to the observed decrease of the photoconversion. It is clear that both the appearance and the extent of this last phenomenon should be strongly influenced by the mixing rates inside the photoreactor and the reactor thickness. Assuming that a 1% conversion variation is already masked by experimental uncertainties, then the loss of 1% of the incoming radiation at the rear wall would not be detected effectively, which means that as soon as the group in parentheses in eq 6 reaches a value of -4.6 a plateau in photoconversion rates is practically reached. The critical catalyst concentration can therefore be estimated as

mcrit =

4.6FpDpR/γ L

(7)

and, as can be seen, it depends on reactor thickness L as well as on catalyst particle size.

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As an example, let us consider the case of commercial TiO2 particles finely dispersed in water solutions. It is known that in this case the TiO2 particles exist in the slurry in the form of aggregates of 0.5-1.0 µm size. Assuming for such aggregates an apparent density of 3000 kg/m3, a spherical shape and a size of 1 µm, and assuming a 3 cm thick reactor, simple substitution of these values into eq 7 gives rise to mcrit ) 0.3 g/L which is in the range of published results for such parameter (Augugliaro et al., 1993). It is worth noting that the model predicts that the amount of lost radiation depends on the value of the product mL, so that by letting this value to remain constant while varying both m and L , the radiation lost through the photoreactor rear wall does not change; this behavior has been experimentally observed (Augugliaro et al., 1991). It is also interesting to note that eq 7 can be regarded as being similar to an equation proposed by Daroux et al. (1985), although their equation is based on more qualitative considerations and on a doubtful estimation of the actual value of the particle size for the particular solid-gas system employed by these authors. If a characteristic extinction length λ0 is defined, eq 6 can be rewritten as

( )

G ) G0 exp -

x λ0

(8)

with λ0 given by

λ0 )

FpDpR/γ m

(9)

and it can be said that the amount of radiant energy which is allowed to leave the photoreactor depends on the number of “lambdas” (L/λ0) that the radiation has to travel inside the reactor before escaping and that about four to five lambdas are sufficient to make the rear wall radiation loss practically negligible. Equation 9 shows that the characteristic extinction length λ0 is simply proportional to the particle size Dp. Consequently, in a given photoreactor with a certain value of the escape length of the photons (distance between photon inlet and outlet, measured in the radiation direction) if the same absorption in the photoreactor is to be maintained while changing the photocatalyst particle size from 1 to 10 µm, the required particles mass concentration m must be increased 10fold, as indicated also by eq 7. It is interesting to observe that under the simplifying assumptions here adopted the rigorous radiation transfer equation (O ¨ zisik, 1973) gives rise to equations practically identical to eq 8, with λ0 replaced by the reverse of the volumetric extinction coefficient (β) which, under the no-scattering hypothesis of the present model, coincides with the volumetric absorption coefficient (κ). Therefore

λ0 )

1 1 ) β κ

(10)

and its value is to be regarded as the photons mean free path in the system before interaction (and absorption) with particles. It is also worthwhile pointing out that the parameter “number of lambdas” (L/λ0) introduced above coincides with the so-called optical thickness of the system.

Finally, the relevant local volumetric rate of energy absorption, here conveniently expressed as [einstein s-1 m-3], is obtained by simply taking the negative derivative of eq 8:

LVREA )

( )

G0 x exp λ0 λ0

(11)

Equation 11 shows that the LVREA also undergoes an exponential decay while moving deeper in the slurry along the radiation direction, and with a decay constant identical to that of the radiation intensity G. The maximum value of the LVREA is attained at the front end of the photoreactor and is given by G0/λ0, i.e., the absorption rate that if it were constant inside the slurry, it would extinguish the entering radiation over a length equal to λ0. When eqs 8 and 11 are compared, it turns out that, at any point inside the photoreactor, the LVREA is simply given by

LVREA ) G/λ0

(12)

Particles with Size Distribution. The case considered so far is that of monosized catalyst particles; however, the extension of the model to the case of particle size distributions is straightforward. Let us consider for instance the case of a bimodal distribution characterized by a concentration m1 of particles with diameter Dp1 and a concentration m2 of particles with diameter Dp2 . Following the previous approach, it is immediately seen that the only change occurs in the expression of Ap which, on the basis of eqs 2 and 5, is given in this case by

m2 Ap m1 ) np1ap1 + np2ap2 ) + ∆x FpDp1R/γ FpDp2R/γ

(13)

Equating eq 13 with eq 5 rewritten with an effective average particle size Dp,av , it is immediately found that the formal solution already obtained (eqs 8 and 9) remains unchanged if a suitable average value for Dp is used, given by

Dp,av )

m1 + m2 m1 m2 + Dp1 Dp2

(14)

By extension, in the case of multimodal size distributions, it can be written that

Dp,av )

mtot mi

∑i D

(15)

pi

Therefore, all equations developed so far, as well as the others going to be obtained, are able to treat also the case of multimodal size distributions by simply utilizing the average particle size defined by eq 15. The extension of eq 15 to the case of continuous size distributions is clearly straightforward and does not deserve further comments. 3. Experimental Section In order to validate the model developed, experimental tests of light absorption were performed. The

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Figure 2. Experimental apparatus.

experimental apparatus used is depicted in Figure 2; it allows the measurement of radiant energy transmitted through solid suspensions in a plane geometry. A collimated portion of the light beam generated by a halogen lamp was filtered by means of an interference band-pass filter (366 nm) and allowed to cross a quasiuniform suspension of solid particles flowing inside the measuring chamber. The exiting beam intensity was measured by means of a photomultiplier after crossing an identical filter in order to minimize the ambient light interference. Visual inspection confirmed that the solid concentration distribution in the measuring chamber could be considered as practically uniform. On the other hand it must be recognised that the solid concentration was not uniform all over the flow circuit, as there is a tendency to accumulate solids in the up-flowing regions while the opposite happens in down-flowing regions. The extent of this effect depends on the values of solid settling rates as compared to the values of average slurry velocity. If the latter are much greater than the former, the effect under discussion is negligible. In a typical experimental run, after the system is filled with clear liquid, the circulation pump was switched on and the amount of light transmitted in these conditions was assessed (GL0). Weighted amounts of solids were then introduced into the system giving rise to step increases of solid particle concentration and the relevant light transmission values (GL) were measured. The solid particle concentration was always computed assuming a uniform solid concentration throughout the system, thus neglecting the abovementioned effect. When the fraction of transmitted light versus solid concentration is plotted on a semilogarithmic plot, the model predicts that a straight line should be obtained (eq 6) whose slope depends only on particle shape and size. Experimental runs were performed with TiO2 particles from three suppliers (Merck, Degussa, and Carlo Erba) in order to test the validity of the model with a typical photocatalyst. As well-known, this photocatalyst is made of extremely tiny particles, which undergo aggregation-disaggregation phenomena in a complex way, depending on a variety of experimental conditions (Martin et al., 1993). For this reason only a qualitative validation of the model was possible by using TiO2 particles. In order to quantitatively test the model, it was thus necessary to resort to other particles. Alumina, activated carbon, zeolite, and other materials were preliminarily tested, but none of them showed a mechanical

resistance that was adequate to obtain meaningful figures. Glass beads were chosen in the end, as they exhibit very good mechanical properties so that no particle comminution was observed under the experimental conditions. On the other hand, their settling velocity is quite high so that the assumption of uniform concentration was not always exactly verified; therefore slight deviations from model predictions were to be expected. It is worth noting that the glass particles utilized were probably unable to completely absorb the UV photons interacting with them. Indeed photons hitting glass particles were probably scattered by reflection and/or refraction as well as being absorbed. On the other hand, in any of these circumstances they were most probably removed from the thin light beam actually employed in the experimental runs. Therefore the great majority of photons that had undergone an interaction with a glass particle was not able to reach the detector anymore, and in this respect these photons actually behaved exactly as if they would have been absorbed upon interaction. It is in light of these considerations that the experimental runs performed using glass particles can be utilized here for model validation purposes. Interestingly, if the cross section of the light beam irradiating the slab, and/or if the size of the detector had been significantly larger than those actually employed, the scattered photons would have been able to significantly contribute to the detector response and the verification of the model, and in particular of eq 6, would have been impossible with fairly transparent particles. In other words the experimental setup employed here actually measures the total extinction coefficient, independent of the fact that upon interaction photons are absorbed or scattered; on the other hand, the extinction coefficient does actually coincide with the absorption coefficient for particles that completely absorb the incident light, and in this respect the glass beads results are suitable for model validation purposes. 4. Results and Discussion As previously mentioned, the experimental results obtained consist of information on the light transmitted through the reactor. For such kind of data the model predicts that a straight line should be obtained when plotting these data against particle concentration, on a semilogarithmic plot, as can be seen when eq 6 is rewritten with x ) L:

ln

( )

GL L m )G0 FpDp(R/γ)

(16)

The slope of the predicted straight lines should thus be related to particle average size by the following equation:

Dp ) -

L Fp(R/γ)(-slope)

(17)

It is worth noting that the experimental measurements actually regarded the radiation transmitted through the whole apparatus, including the front and rear glass walls, rather than the radiation entering and leaving the catalyst suspension. Let us call Gfw, Grw, and Grw0 respectively the photon flux entering the front glass wall, that leaving the rear wall, and that leaving the rear wall when m ) 0. Indicating also with τfw , τb, and τrw the transmittances of the front wall, those of the suspension slab, and those of the rear wall, it is evident

Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4745

Figure 3. Experimental results for TiO2 particles.

that G0 ) τfwGfw, GL ) τbG0 ) τfwτbGfw, and Grw ) τrwGL ) τfwτbτrwGfw. When m ) 0, τb ) 1; thus Grw0 ) τfwτrwGfw and therefore (Grw/Grw0) ) τb ) (GL/G0). In other words the ratio (Grw / Grw0), the one that was actually measured, can replace the ratio (GL / G0) in all relationships, and this will implicitly be done in all the following. TiO2 Results. Typical results are shown in Figure 3 where it can be observed that straight lines are actually obtained, as predicted by the model, resulting in a qualitative confirmation of the validity of the model in this case. The slopes of these straight lines fitting the data are -6.2, -35, and -95 m3 kg-1 for Merck, Degussa, and Carlo Erba TiO2, respectively. As already mentioned, the particle size and shape were not known in this case so that a quantitative validation of the model could not be obtained. On the other hand the particle size can be estimated by means of the experimentally observed slopes and eq 17. By assuming a spherical shape (R/γ ) 2/3) and an apparent density of the aggregates of 3000 kg m-3 and applying eq 17, the resulting particle sizes are 4.03, 0.71, and 0.26 µm for Merck, Degussa, and Carlo Erba TiO2, respectively. As can be seen for Degussa P25 and especially for Carlo Erba TiO2 the calculated particle size is so small that one of the main hypotheses (assumption 1) of the model is not fulfilled. This means that the real particle size may differ somehow from the estimated value, but it is interesting to observe that one of the main features of the model (exponential dependence on solid concentration) is confirmed also in these extreme cases. It is worth noting that during the runs, especially with Degussa P25 particles, after a new amount of solids is added to the system, the measured value of transmittance showed a tendency to increase in time and only after several minutes settled on steady-state values, which are those reported in Figure 3. This behavior was attributed to the fact that the TiO2 particles utilized form aggregates in the slurry, whose size depends on many parameters including mechanical interactions in the apparatus. As the light transmission characteristics depend on the size of these aggregates, the observed time dependence of transmitted light can be explained assuming that the formation and stabilization of the aggregates takes some time. It is finally worth noting that no strong influence of the solution pH was observed in the case of Degussa

Figure 4. Experimental results for glass beads (Dp ) 90-105 µm).

P25 particles. This result is somewhat different from that observed by Martin et al. (1993), who found an influence of pH on the average size of the aggregates. This discrepancy might be explained on the basis of the fundamentally different conditions of mechanical stress and the poorer purity of the system in the present case. Glass Beads Results. Experiments were performed with particles of several diameters (40-70, 90-105, 125-150, 150-180, and 210-250 µm). Obviously in this case no time dependence of the transmittance was observed. Typical results are shown in Figure 4, which was obtained with sieved glass beads with sizes ranging from 90 to 105 µm. It can be observed there that a straight line is obtained again, with a slope of -0.27 m3 kg-1. If a particle density of 2600 kg/m3 is assumed, a spherical shape and a photon escape path of 5 cm, a computed average particle size of 106.8 µm is obtained, which compares fairly well with the actual size range of the particles. Results similar to those shown in Figure 4 were obtained with all the other glass beads tested. The values of the experimentally observed slopes were converted by means of eq 17 in the corresponding particle diameters, resulting in the following values for the “computed” average diameter 77.8, 106.8, 144.0,

4746 Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997

fore also this last equation receives an experimental confirmation. 5. Conclusions

Figure 5. Experimental versus computed values of Dp.

Figure 6. Experimental results for glass beads (50% w/w mixture of 90-105 and 212-250 µm).

154.6, and 192.0 µm for the above-mentioned five size ranges, respectively. These values are plotted in Figure 5 versus the arithmetic mean of the particle size range. A scrutiny of Figure 5 shows that a general satisfactory agreement exists between computed and actual particle sizes. It may be observed that the relatively small deviations from the diagonal show a systematic trend. This last feature can be explained in view of the alreadymentioned oversimplifications of the model with respect to the actual experimental conditions, and in particular (i) the fact that the glass particle transparency was neglected and (ii) the lack of uniformity of particle concentration in the system, which increases with particle sedimentation speed and therefore with particle size. In any case it can be stated that, at least for approximate calculations, the model is to be considered as being quantitatively validated. In order to test the reliability of the model in the case of catalyst particles with size distributions, an experimental run was performed in which a 50% w/w mixture of 90-105 and 212-250 µm glass beads was used as the solid phase. The results of this run are plotted in Figure 6 where it can be seen that again a straight line fits the data quite well. The slope of this line (-0.21 m3 kg-1) introduced into eq 17 leads to an estimated average size of 137.4 µm which compares very well with that (137.1 µm) expected on the basis of eq 15. There-

A very simple model (ZRM) for the prediction of radiation fields inside heterogeneous photocatalytic reactors has been devised, for the case of completely transparent fluid phases and of “big” catalyst particles that do not reflect or scatter photons of suitable wavelength. For these cases, the ZRM model allows estimation of the extinction characteristics of the slurry once the particles concentration and size distribution are known, thus making it possible to carry out fully predictive simulations. Although experimental limitations, mainly related to the unavailability of strong “black” particles of known size, did not allow the collection of fully relevant data for the quantitative validation of the model, the qualitative results obtained with TiO2 particles together with the results obtained with glass beads and with the simple geometrical considerations on which the model is based, lead to the conclusion that the ZRM model can be considered as being quantitatively validated. The main feature of the ZRM model is its simplicity, that allows one to get an immediate grasp of the way in which the main physical parameters affect the photoreactor performance, which is often all that is needed for engineering purposes. For instance it is immediately seen that, in order to maintain the same radiation profiles in a given photoreactor, when doubling the particle size, the solid concentration in the system must be doubled as well (see eq 6). The model also allows an immediate estimation of the required catalyst concentration needed to practically extinguish the radiation in a reactor of any given thickness (eq 7), once the particles size is either known or has been assessed on a different photoreactor on the basis of eq 17. Obviously the ZRM model is formally applicable to physical situations in which all the model hypotheses are strictly abode by. It can also be confidently applied to situations that differ somewhat from the ideal case, accepting that only approximate results will be obtained in this case, as is customarily done for engineering purposes. In particular it can be confidently applied to photocatalysts that are able to effectively absorb the photons carrying an energy greater than the band gap, i.e., that upon interaction scatter only a minor proportion of the useful photons. The case of photocatalysts that scatter greater proportions of interacting photons is the object of Part 2 of the present paper, where a quantitative assessment of the approximations incurred by the ZRM model, for cases in which a noticeable scattering exists, will also be discussed. Acknowledgment This research work has been carried out under a financial support of the STEP Research Programme of E.E.C. (Contract no CT91-0133.) Nomenclature ap ) projected area of one particle [m2] Ap ) sum of all projected areas of particles statistically present in a volume of thickness ∆x [m2] Dp ) catalyst particle diameter [m] Dp,av ) average particle diameter (eq 15) [m] Dpi ) ith particle diameter [m] G ) average forward radiant flux [einstein s-1 m-2]

Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4747 G0 ) average radiant flux at locations x ) 0 and r ) ro [einstein s-1 m-2] L ) photoreactor thickness in the radiant flux direction [m] LVREA ) local volumetric rate of energy absorption [einstein s-1 m-3] m ) catalyst concentration [kg m-3] mcrit ) “critical” catalyst concentration [kg m-3] mi ) catalyst concentration of ith particle diameter [kg m-3] mtot ) total catalyst concentration [kg m-3] np ) number of particles per unit volume [m-3] vp ) volume of one particle [m3] x ) distance from photoreactor front plane [m] Greek Symbols R ) shape factor for particle volume (particle volume ) RDp3) β ) volumetric extinction coefficient [m-1] γ ) shape factor for particle projected area (particle projected area ) γDp2) κ ) volumetric absorption coefficient [m-1] λ0 ) characteristic extinction length or photons mean free path [m] Fp ) catalyst density [kg m-3]

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Received for review May 8, 1996 Revised manuscript received November 15, 1996 Accepted June 16, 1997X IE960259J

X Abstract published in Advance ACS Abstracts, August 15, 1997.