Simplified Modeling of Radiant Fields in Heterogeneous

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Ind. Eng. Chem. Res. 1997, 36, 4748-4755

Simplified Modeling of Radiant Fields in Heterogeneous Photoreactors. 2. Limiting “Two-Flux” Model for the Case of Reflectance Greater Than Zero 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

In the first part of this work a simple model for the description of the radiant field in heterogeneous photoreactors was developed, based on the assumption of zero reflectance of the particles. In this second part of the investigation a limiting model is developed in which the above hypothesis is removed. In this last model, scattering phenomena are dealt with in a very simple “two-flux” way, so that analytical solutions are obtained again. The case here developed is the same as in part 1, namely slab geometry with orthogonal parallel irradiation, giving special reference to the important case of semi-infinite reactor thickness. Both cases of solid catalysts with single particle size and with distributed particle sizes are implicitly dealt with. The model here developed represents a limiting case as it overestimates the radiation lost from the reactor, giving rise to an opposite behavior with respect to the predictions given by the simpler model developed in part 1. Therefore, the real behavior of heterogeneous photoreactors lies between the predictions of the two models. It is also shown that for a fairly wide range of reflectance values, the predictions of the two models do not differ significantly. 1. Introduction In the “zero reflectance model” (ZRM) developed in part 1 of this paper (Brucato and Rizzuti, 1997), it was assumed that, upon interaction with solid particles, the probability of a photon being scattered (rather than being absorbed) is zero. If the real value of this probability is significantly different from zero, the ZRM clearly underestimates the number of photons actually leaving the system from both the front and rear walls, thus resulting in an overestimation of the number of photons absorbed in the photoreactor. An accurate radiant field model for this case has to account for the scattered light contribution to light intensity at each point in the photoreactor. Information on the “phase function” describing the directional distribution of scattered energy is needed (O ¨ zisik, 1973), and a directional radiation transfer equation (RTE) has to be solved. Even in the simple case of plane geometry and parallel irradiation of the photoreactor, analytical solutions cannot be obtained any more with realistic phase functions. Only numerical solutions are attainable in this case, and these require considerable computational efforts due to the integro-differential nature of the RTE (Santarelli, 1985; Alfano et al., 1986; Cassano et al., 1995; Brandi et al., 1996). The task is somewhat simplified by the adoption of random walk Monte-Carlo approaches, which have proved to be quite viable for solving the problem (Spadoni et al., 1978; Pasquali et al., 1996). A different approach is pursued here; simplifications are made in such a way that the following features are met: (i) an analytical solution can be obtained; (ii) the model results in an underestimation (as opposed to the overestimation performed by the ZRM) of the number of photons actually absorbed in the photoreactor. In this way, the present model predictions will lie on the other side, with respect to reality, from those of ZRM. A comparison of results predicted by both models will then * To whom correspondence should be addressed. E-mail address: [email protected]. S0888-5885(96)00260-6 CCC: $14.00

allow an evaluation of the lack of realism associated with their use. 2. Simplified Two-Flux Model (TFM) When a photon hits an opaque solid particle, it can be either absorbed or scattered in various directions. If the particle is an opaque sphere and its surface is scattering in a diffuse manner (i.e., if it is rough enough to locally give rise to a “spherical” scattering, as is actually the case for most photocatalyst particles), it is well-known that the relevant phase function Φ(φ), i.e., the probability of being scattered in a direction forming an angle φ with the incident direction, is given by (Siegel and Howell, 1992)

Φ(φ) )

8 (sin φ - φ cos φ) 3π

(1)

Equation 1 shows that a scattered photon has the maximum probability of being scattered in the “pure backward” direction (φ ) 180°), a zero probability of being scattered in the “pure forward” direction (φ ) 0°), and significant probabilities of being scattered in any other direction including those having a component in the forward direction (φ < 90° and φ > 270°). For modeling purposes, in the following it will be assumed that when a photon is reflected by a particle, only “pure back-scattering” occurs (i.e., φ ) 180° always). This is clearly a hypothesis that, as in the case of the ZRM, oversimplifies the reality. Nevertheless, it leads to a model that satisfies both the above mentioned requirements, i.e., that can be analytically integrated and that underestimates the number of absorbed photons with respect to the real behavior of the solid dispersion. In order to understand this last statement, one can consider that a photon that is scattered with φ ) 180° has to travel the minimum distance in the system before escaping from the front end of the plane photoreactor: thus, it has the minimum chance of hitting another particle (being then either absorbed or scattered again). © 1997 American Chemical Society

Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4749

x and x + ∆x, results in

Gx+∆x ) Gx - G(1 - R)npap∆x out in absorbed gRnpap∆x GRnpap∆x + (3) back-scattered “out” back-scattered “in” from which the following differential equation is obtained:

dG ) -npap(G - gR) dx Figure 1. Schematic representation of photon fluxes in the slab photoreactor.

All photons scattered in any other direction will have greater chances of undergoing another particle encounter and thus greater chances of being absorbed. Now it is clear that the assumption that all scattered photons are purely back-scattered implies an underestimation of the amount of photons absorbed in the system. On the other hand, the main assumption of ZRM (zero probability for a photon to be reflected) implies an overestimation of the same quantity. Therefore, it can be stated that the two models give rise to predictions that bound the real photoreactor behavior. Model Assumptions. Summarizing, as regards the assumptions of this second model, the first and the second of the assumptions made in the ZRM model developed in part 1 of this paper (Brucato and Rizzuti, 1997) are retained: (1) the catalyst particle size is significantly greater than the photon wavelength; (2) only two types of events can result from the interaction of radiation and particles: absorption and/or reflection. The third assumption (no reflection upon interaction) is replaced by the following: (3) when a photon hits a particle, it has a probability R of being scattered (0 < R < 1) and in such an event it is purely back-scattered (φ ) 180°). Other hypotheses, usually verified in photocatalytic reactors, are complete transparency of the liquid phase to the photons of interest, negligible emission of both the liquid and the solid phase, “elastic” scattering only (no change of photon wavelength upon interaction), and absence of scattering centers different from the photocatalyst particles. It is worth noting that, due to its definition, the model parameter R practically coincides with the “single scattering albedo”, i.e., with the ratio of the “volumetric scattering coefficient” σ and the “volumetric extinction coefficient” β ) κ + σ (where κ is the “volumetric absorption coefficient”), often employed in the field of radiation transfer in participating media (O ¨ zisik, 1973):

R)

σ σ ) β κ+σ

(2)

As a consequence of the third hypothesis, in the case of parallel irradiation of the photoreactor, only two directions for light propagation are to be considered, as the possibility for a photon to travel in any other direction is not admitted by the model. This model thus belongs to the class of the “two-flux models” (O ¨ zisik, 1973) recently employed for heterogeneous photochemical reactor modeling purposes (Maruyama and Nishimoto, 1992). In the following it will be referred to as TFM. With reference to Figure 1, in the present case there will be two different photon fluxes: a forward flux (G), and a backward flux (g). A photon flux balance applied to the forward flux in the differential volume between

(4)

The same procedure can be applied to the backward flux, resulting in

dg ) npap(g - GR) dx

(5)

Equations 4 and 5 give rise to a system of two ordinary differential equations in the two unknown functions G(x) and g(x) that has to be solved by taking into account the proper boundary conditions. In the present case, neglecting the reflection phenomena at front and rear walls, the following boundary conditions have been used:

B.C.1 B.C.2

G ) Go g)0

for for

x)0

(6a)

x)L

(6b)

The statement of the problem is somewhat simpler here than that reported by Maruyama and Nishimoto (1992), although the main assumptions appear to be the same. Please note that the observations discussed in part 1 of this paper (Brucato and Rizzuti, 1997) concerning the deviations from the simple nonshading of particles inside the same ∆x obviously apply to this two-flux case as well. The application range of the results is thus limited again to the case of fairly diluted suspensions. On the other hand, the solution has a simple structure and makes for an easy physical understanding of the effect of the main parameters. To solve the system, g is obtained from eq 4 and substituted into eq 5, yielding

d2G ) np2ap2(1 - R2)G dx2

(7)

Equation 7 is a second-order, linear, homogeneous differential equation that can be effortlessly integrated. The two integration constants are then computed by imposing the following boundary conditions:

B.C.1′ B.C.2′

G ) Go

for

-dG/dx ) npapGL

x)0 for

(8)

x ) L (9)

where B.C.2′ was obtained from eqs 4 and 6b. In this way one obtains

G)

Go (e-x/λR - ηex/λR) 1-η

(10)

where η is defined as

η)

1 - x1 - R2 -2L/λR e 1 + x1 - R2

(11)

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while λR is defined as

λR )

1

(12)

npapx1 - R2

λR has dimensions of length and is analogous to the “characteristic extinction length” λo introduced in eq 9 of part 1 (Brucato and Rizzuti, 1997). This last was shown to be related to particle shape, size, and concentration, so that it can now be written

λR )

λo

where

x1 - R2

λo )

FpDp avR/γ (13) m

which holds true in the case of single size catalyst particles as well as in the case of particles with size distribution. In this last case it is sufficient, in fact, to compute a convenient average size Dp av, as indicated by eq 15 in part 1 (Brucato and Rizzuti, 1997). It is worth noting that while λo coincides with the ratio 1/β and has the physical meaning of the photon mean free path in the system before interaction with particles (Brucato and Rizzuti, 1997), no such fundamental meaning can be given for λR. Nevertheless, it will be shown in the following that, for practical purposes, λR can be regarded as the “characteristic extinction length” for the cases where R > 0. By substituting eq 10 into eq 4, the backward photon flux is obtained:

g)

Figure 2. Radiant fluxes for the case of R ) 0.5 and L ) λo.

Go

[(1 - x1 - R2)e-x/λR - η(1 + R(1 - η)

x1 - R2)ex/λ ] R

(14)

It is interesting to note that, differing from the case of zero reflectance, the values of both radiant fluxes at a generic location depend on the photoreactor total thickness L, which is involved in the value of η. This is not surprising since the thicker the photoreactor, the greater the backward radiation as it receives contributions from more layers. Moreover, as L increases, the forward radiation also increases because it receives more contributions from that part of the backward radiation reflected in the forward direction by scattering events. It can be observed that in the case of zero reflectance (R ) 0), η is equal to zero (eq 11) and thus eq 10 correctly reduces to eq 8 of part 1 while eq 14 gives g ) 0 everywhere, as expected. It should be noted that for the cases in which R is significantly different from zero and a polychromatic irradiation exists, the present analysis must be applied separately either to each wavelength of interest using the relevant value of R or to suitably discretized wavelength ranges, like Brandi et al. (1996). A photon balance on the small volume between x and x + ∆x shows that the local volumetric rate of energy absorption (LVREA) is simply given by

LVREA ) -

dG dg d(g - G) + ) dx dx dx

(15)

By substituting eqs 10 and 14 into eq 15, one obtains

LVREA )

Go

[(R - 1 + x1 - R2)e-x/λR + η

λRR(1 - η)

(R - 1 - x1 - R2)ex/λR] (16)

Figure 3. Radiant flux reflected by the plane slab.

In Figure 2 the model predictions for the case of R ) 0.5 and L ) λo are shown. Both G and g show a decreasing trend when going deeper into the reactor, as could be expected. On the same figure, the values of G predicted in the case R ) 0 are also shown as a dotted line for comparison purposes. It can be observed that in spite of the fairly large difference of albedo values (R ) 0 and R ) 0.5, respectively), the predicted curves for G do not differ very much and therefore also the fraction of radiant flux lost at the rear wall of the slab (GL/Go) is almost the same. The value of g at the front end of the reactor (go) represents the radiant flux in the backward direction lost at the reactor front end (i.e., the radiant flux “reflected” by the slab), which for the particular case considered in Figure 2 is equal to 22.3% of the incoming radiation Go. This “reflected” radiant flux obviously depends on both slab thickness L and albedo R. Its expression is immediately obtained from eq 14 by letting x ) 0:

go )

Go 1+η x1 - R2 1R 1-η

(

)

(17)

where the first factor shows the main dependence of go on R while the dependence of go on the slab thickness L is embedded in the parameter η (eq 11). The value of the fraction of incoming radiation lost by reflection (go/Go) obtained from eq 17 for various values of R, is represented in Figure 3 where it is plotted

Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4751

Figure 5. Transmitted radiant flux versus L/λo.

Figure 4. Radiant flux transmitted through the slab.

versus the ratio L/λR. It can be observed there that the value of go approaches its asymptotic value already for values of L/λR in the range from 2 to 3. In other words, the amount of radiation lost at the front wall of the photoreactor practically reaches quite soon its maximum value, so that relatively thin photoreactors do behave in this respect just as if they were infinitely thick. This result is not surprising if one considers that the only way in which the slab thickness affects go is through the value of η. Due to its definition (eq 11), the value of η is always smaller than 1 and rapidly vanishes due to the presence of the exponential term. Moreover, this last term depends on -2(L/λR) so that it becomes negligible already for L/λR values between 2 and 3. The other local quantity of interest is the amount of forward radiant flux lost at the rear wall of the slab, GL, which is also a function only of R and L/λR. By letting x ) L into eq 10 and partially substituting the expression of η, the expression for GL can be put in the form

GL )

(

1 1-η

2x1 - R2

1 + x1 - R

2

)

Goe-L/λR

the hypothesis of R ) 0 would result in a different behavior under this respect. Figure 4 shows that curved lines are actually obtained in the case of R > 0, but deviations from a straight line trend would be masked by the inevitable experimental scatter of data in most cases, unless the value of R is fairly close to 1. In this last case, as can be observed in Figure 4, distinctly curved lines should be obtained, which practically settle on straight lines for (L/λR) > 2. However, as a difference from the cases of R not close to 1, the intercept of this straight line with the ordinate axis is not 1, but a smaller value (ψ) dependent on R. As an example, the intercept ψ pertaining to the case R ) 0.99 is indicated in Figure 4. This behavior is easily understood by looking at eq 18. It has already been observed in fact that the value of η vanishes quite soon when L/λR increases so that it quickly becomes negligible in comparison with 1 and can be removed from eq 18. From then on the only dependence of GL/Go on L/λR reduces to the exponential term, and this gives rise to the straight line portion of the curves. The relationship between R and the intercept of this straight line with the ordinate axis is now clear: it is simply given by the expression in parentheses in eq 18.

(18)

A semilog plot of GL/Go versus L/λR is reported in Figure 4 where it can be observed that up to fairly high values of albedo (e.g., R ) 0.5), the difference with the zero reflectance case is negligible, as practically the same straight line is obtained. Only for values of R approaching 1 does the difference become important and are curved lines obtained. This is an interesting result in view of the fact that, for a given catalyst (i.e., for fixed values of R, Fp, R/γ, and Dp) and for a given photoreactor (fixed value of L), varying the catalyst concentration (m) results in a proportional variation of the ratio L/λR, as it is immediately seen by recalling the dependence of λR on m (eq 13). Therefore, the plots in Figure 4 are representative of what would be obtained if results from a conventional experiment on the dependence of light transmittance on catalyst concentration were plotted versus this last parameter. In particular, it was demonstrated in part 1 (Brucato and Rizzuti, 1997) that in the case of R ) 0, straight lines are obtained in this kind of plots and one may wonder if any deviation from

Ψ)

2x1 - R2 1 + x1 - R2

(19)

The structure of this equation is such that significant deviations of Ψ from the unity value occur only at relatively large values of R. In looking at Figure 4, it might appear surprising that the curves pertaining to greater values of albedo lie underneath those pertaining to smaller values of the same parameter, as if it occurred that the greater the albedo, the smaller the transmitted flux. The reason for this apparently contradictory behavior is that, due to the definition of λR (eqs 12 and 13), the actual length scale of the abscissa changes when changing R. If the same ordinates are plotted versus L/λo, thus making the length scale independent of R, Figure 5 is obtained, where the greater the value of R, the greater the value of GL at equal L/λo, in agreement with what could be expected. The curves shown in Figure 5 are representative, for instance, of what should be obtained with a given catalyst type in a given slab, by varying the catalyst concentration (hence λo) in order to obtain each

4752 Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997

curve, and changing the curve parameter R by simply changing the radiation wavelength. Limiting Case of R ) 1. This limiting case might be considered uninteresting for photoreactor modeling, as catalyst particles do have to absorb radiation in order to promote photoprocesses. However, it is still interesting not only as a limiting case but also in order to understand what occurs to photons of energy smaller than that required for the photoprocess promotion. This is, for instance, the case of practically all visible radiation (wavelength greater than 390 nm) when TiO2 is used as a photocatalyst. As a matter of fact, in this range the albedo of TiO2 is constant and practically equal to 1 (Alfano et al., 1994). It has just been shown (Figure 5) that in the case of R ) 1, the radiant energy transmitted though the slab (GL) decreases while L/λo increases, i.e., while the particle concentration and/or slab thickness increases. In order to better understand the behavior observed in Figure 5, the analytical expression for GL can be obtained from eq 18 by letting R tend to 1 and applying De L’Hopital’s rule to the resulting indefinite form. In this way one obtains

GL )

1 G 1 + L/λo o

(R ) 1)

eqs 10 and 14, resulting in the following equations, respectively

G ) Goe-x/λR

(20) g)

which shows that in this case the relationship between (GL/Go) and (L/λo) has a hyperbolic rather than an exponential nature, so that it never settles on a straight line in semilog plots. The same procedure applied to eq 17 leads to

go 1 )1Go 1 + L/λo

Figure 6. Radiant flux reflected by a semi-infinite slab.

(R ) 1)

(21)

which shows that the radiant flux reflected by the slab also has a hyperbolic structure. Moreover, for any value of L/λo it is (GL + go) ) Go, and therefore no radiation is absorbed in the slab, as could be expected by observing that in the case of R ) 1 the LVREA must be equal to zero everywhere. In other words, when the slab thickness (or particle concentration) is increased, simply a greater portion of Go is reflected rather than being transmitted. The structure of eqs 20 and 21 is so simple that it is not worth plotting them on a graph. Let us just observe that a value of 9 for L/λo is required for reducing to onetenth the transmitted energy and a value of 99 is required in order to reduce it to one-hundredth. In the case of a semi-infinite slab (L/λo f ∞), no radiation can be transmitted at all and the radiation is entirely reflected (go ) Go). Case of a Semi-infinite Slab. The case of a semiinfinite slab is particularly significant, as in commercial photoreactors a negligible amount of potentially useful radiation would be allowed to escape from the system. Thus, the reactor thickness will amost certainly be sufficiently large to obtain this result (e.g., L > 4-5λR), and it can be expected that the numerical differences between this case and the ideal case of a semi-infinite slab will be practically negligible. In the following such photoreactors will be referred to as “thick” photoreactors. When L/λR tends to infinity, the parameter η tends to zero (eq 11). Therefore, the expressions for G and g in this case are simply obtained by allowing η ) 0 into

(L/λR ) ∞)

1 - x(1 - R2) G R

(L/λR ) ∞)

(22) (23)

which show that, in this significant case, the direct radiation G exhibits a simple exponential decrease while going deeper inside the photoreactor, exactly as in the case of zero reflectance (eq 8, part 1), but with the greater characteristic length given by eq 13, as previously stated. Interestingly, the backward radiation flux g shows the same exponential decrease as the forward radiant flux, and their ratio is constant at any point inside the photoreactor (eq 23). As regards the practical applicability of eqs 22 and 23, it can be immediately assessed that for “thick” photoreactors, even in the case of large values of the albedo R, the differences between the radiant fluxes predicted by eqs 10 and 14, and by the simpler eqs 22 and 23, are practically negligible. As a matter of fact, large differences can be observed only near the rear wall of the reactor, but the same differences are negligible when compared with Go, in view of the small absolute values of G at the rear wall of the slab. As the overall performance of the photoreactor depends on the fraction of Go absorbed in the slab reactor, it can be concluded that for practical purposes eqs 22 and 23 can be employed instead of eqs 10 and 14, for relatively thick photoreactors (L > 4-5λR), with very little loss of overall accuracy. For instance, it can be easily calculated that for the case R ) 0.9 and L/λR ) 5, the overall absorbed radiation (Go - GL - go) is 36.91% of Go for the “thick” photoreactor (eqs 10 and 14), while it would have been estimated as 36.65% of Go if the equations pertaining to the semi-infinite photoreactor had been used instead. Some other relationships can benefit from the simplicity of the solution for the semi-infinite case. For instance, the energy lost by backward radiation at the front wall of a semi-infinite photoreactor is immediately obtained by the ratio g/G (eq 23):

go 1 - x1 - R2 ) Go R

(L/λR ) ∞)

(24)

Equation 24 is reported in Figure 6 as go/Go versus R. It is interesting to observe that for R < 0.5 the

Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997 4753

Figure 7. LVREA for a semi-infinite slab and various R.

numerical value of the fraction of incoming radiation lost at the front wall practically coincides with R/2. This feature is easily explained by substituting for

x1 - R2

its approximate expression (1 - R2/2) obtained by series expansion of the root, truncated at the second term:

go R ≈ Go 2

(L/λR ) ∞ and R < 0.5)

(25)

The simplicity of this last result gives an easy-toremember rule of thumb for the estimation of the magnitude of the actual fraction of energy lost as a consequence of scattering; it should also be kept in mind that this is anyway an overestimation of the real quantity, so that it can be stated that as long as the albedo of the solid phase is reasonably small, the loss of energy due to the scattering effect may be either neglected or roughly (over)estimated by means of eq 25. Finally, the local volumetric rate of energy absorption for the case of a semi-infinite slab is easily obtained by letting η ) 0 in eq 16:

LVREAR>0,s-inf slab )

1-R G (1 + R - x1 - R2) (26) R λo

which shows that the relevant LVREA has the same exponential decrease as G, to which it is related by a simple proportionality, as in the case of the ZRM (eq 11 of part 1, to which eq 26 correctly reduces in the case of R f 0), but with a correction factor depending on R. The LVREA profiles predicted by eq 26 have been plotted in Figure 7, for several values of R, versus x/λo (in order not to change the length scale when changing the value of R). It can be observed that, in the first part of the slab, the greater the value of R, the smaller the value of LVREA, as could be expected. On the contrary, for x/λo greater than about 3, the greater penetration of photons into the slab results in greater values of LVREA for greater albedo values; it is worth noting that a similar behavior (LVREA profiles crossing) was obtained also by Alfano et al. (1994) with a more rigorous numerical solution of the RTE. Interestingly, for an albedo value of R ) 0.2 the maximum deviation of the relevant profile with respect to that predicted by the ZRM model (R ) 0) is about 10%. If one considers that, as already stated, the real

LVREA profile will certainly lie between the two curves, it becomes evident that the predictions of the simple R ) 0 model can be utilized with good approximation also for all cases of relatively small albedo values. Comparison of TFM with ZRM Predictions. In the considerations developed so far it was noticed sometime that, provided that R was small enough, no great differences existed between the predictions of TFM and ZRM. This section is aimed at pointing out such differences and identifying the field of R values where (and how) the two simplified modeling approaches can be used. Obviously, the comparison can be performed for all the radiant fluxes discussed so far and the simplicity of the relevant equations developed makes it very simple. There is no doubt, however, that for photoreactor design purposes the most important quantity to look at is the LVREA. Therefore, for the sake of shortness, the comparison between TFM and ZRM will be limited here to the LVREA predictions. It can be observed that at all values of R the maximum percent difference between the LVREA predicted by the TFM and that predicted by the ZRM is always found at the front wall of the photoreactor. In particular for R ) 0.2 this maximum difference is about 12%. Considering that the real behavior will lie between the ZRM and TFM predictions, the maximum error incurred by using the simpler ZRM will probably be in the range of 5-6% and the average error will certainly be less than that. Obviously, for smaller values of R the corresponding errors will be even smaller. The following can be concluded: (a) Up to R values of 0.2, the ZRM is the recommended model, as full advantage can be taken from its extreme simplicity, without significant losses of accuracy. (b) For greater values of R, up to, e.g., R ) 0.5, it will probably be sufficient to compute the LVREA with both models and take the average value: the errors incurred will probably be quite small (probably of the order of (10% for R ) 0.5), and in any case the uncertainty range will be known with precision. An alternative choice may be that of selecting the TFM and using it knowing that, as an underestimation of radiant energy absorption is actually made, the resulting design will be conservative. (c) For even greater albedo values (R > 0.5) the differences between ZRM and TFM predictions, and therefore the uncertainty range, become so large that the ZRM-TFM couple should not be used for quantitative computations. Nevertheless, both models still retain an interest for the qualitative interpretation of experimental results as well as for the semiquantitative prediction of the way in which varying the main parameters (e.g., Dp, m, L) may affect the photoreactor performance. In any case, when using the TFM the simpler equations for semi-infinite slabs should be the preferred ones, as they have a simplicity comparable with that of ZRM and the loss of accuracy, with reasonably “thick” beds, will certainly be negligible in front of the errors arising anyway from all other sources of uncertainty. Comparison with Literature Data. Although the “lost” radiant fluxes, GL and go may, at least in theory, be experimentally accessed, it should not be forgotten that the TFM here developed is to be regarded as a “limiting” model that certainly overestimates these quantities, so that a direct comparison of experimental data with model predictions would not add much significance to the present work. This, together with the fact that a proper determination of GL and go is a

4754 Ind. Eng. Chem. Res., Vol. 36, No. 11, 1997

very difficult task, is actually the main reason why an experimental validation of TFM was not attempted here. On the other hand, the ZRM developed in part 1 of this paper (Brucato and Rizzuti, 1996), being a descriptive model, was experimentally validated and it should be recognized that this means also validating that part of TFM that is common with the ZRM, namely the quantification of the probability of interaction between photons and “big” particles (i.e., the link between λo and particle’s characteristics, eq 13). In any case the scientific literature reports some data that may give at least a qualitative confirmation of some of the features predicted by the TFM. First of all, a value for the single parameter in which scattering phenomena have been forced (R), in the wavelength region of interest for the photoprocess promotion, has to be assessed. Concentrating the attention on the TiO2, the most used photocatalyst, it can be said that there is no abundance of data on its optical behavior. Information can be found in a recent paper by Alfano et al. (1994). These authors, by fitting model parameters to experimental data, obtained a table of values of κ and σ for Aldrich TiO2, as a function of photon wavelength. From this table R can be calculated by means of eq 2. It can be observed that at wavelengths shorter than 360 nm, i.e., in the range of interest, R takes surprisingly high values of about 0.9. Therefore, it should be expected that with that TiO2, as well as probably with TiO2 from other suppliers, transmittance measurements should approximately exhibit the behavior of the relevant curve (R ) 0.9) in Figure 4, with an intercept of the straight portion on the ordinate axis Ψ ≈ 0.61. Actually, one has to remember that the real behavior of catalyst suspensions should lie, in all respects, between the predictions of the TFM and those of the ZRM, and therefore the intercept Ψ should be expected to lie between Ψ ) 0.61 (TFM) and Ψ ) 1.0 (ZRM). This is what was actually found by Augugliaro et al. (1995), who performed a series of experiments in which the transmittance of a plane photoreactor was measured by an actinometric method sensitive only to the wavelengths of interest. These authors worked with constant concentration (1 kg/m3) suspensions of “home-prepared” TiO2 particles and measured the light transmitted by plane slabs of different thicknesses. Therefore, in this case the ratio L/λR was varied by varying L while λR remained constant for each of the particle sizes tested. They found that the results could be fitted by a straight line in a semilog plot, whose intercept with the ordinate axis was about 0.75 for all the TiO2 particles tested, thus in agreement with the above stated expectations. The curved portion of the line was not observed, but their experimental setup inhibited the collection of data at small values of L. It is worth noting that their interpretation of this result as an indication that the radiation reflected by the suspension coincides with (1 - Ψ), thus being 25% in this particular case, is not supported by the present model. It is worth recalling that a similar behavior was not observed in the light transmittance experiments described in part 1 of this paper (Brucato and Rizzuti, 1997). On the contrary, with all the TiO2 particles tested, straight lines were obtained on semilog plots, which intercepted the ordinate axis at 1.0 (as if the TiO2 was characterised by R values significantly smaller than 1). On the other hand, it was clearly stated in part 1 (Brucato and Rizzuti, 1997) that, due to the setup arrangement, little scattered radiation was allowed to

reach the detector so that the whole apparatus was suited to measure the single scattering extinction coefficient of the particles suspension rather than the true transmitted radiant flux. This feature was a useful characteristic that allowed the validation of the ZRM even by using partially transparent glass particles, but the same feature makes the TiO2 transmittance data collected meaningless for TFM qualitative validation purposes. This feature might explain the discrepancy with the previously discussed literature data. Other data have been reported by Martin et al. (1993), who found that changes in particle size result in drastic changes of the optical behavior of TiO2 suspensions. They also found that an apparent Lambert-Beer law holds when the concentration and/or path length changes, a result that would be in agreement with TFM predictions for values of R smaller than the 0.9 previously hypothesized. On the other hand they state that in their experimental setup single scattering extinction coefficient, rather than true transmittance, was probably measured. These data would then result again in a validation of ZRM, but no conclusion could be drawn for TFM. It is finally worth noting that Alfano et al. (1994) presented a table of κ and σ values for TiO2 particles of several sizes, computed on the basis of the Mie theory, that lead to R values of about 0.35. This is a quite small value that would completely change the expected behavior on the basis of TFM predictions. It can be concluded that, although the above considerations cannot certainly be considered, for the TFM model here proposed, as a quantitative validation (which would be impossible anyway due to the “limiting” nature of the model itself), there are indications that the TFM can at least provide a framework that can help understanding the real behavior of photocatalyst suspensions. It is also evident that there is a need for performing more experimental work on TiO2 suspensions, before firm conclusions on their optical properties can be drawn. 3. Conclusions A radiant energy transfer model has been developed that, by oversimplifying the scattering phenomena in photocatalyst particle suspensions, allows analytical solutions to be obtained, a particularly interesting feature from the engineering point of view, especially in consideration of the simplicity of most of the equations obtained. The model parameters have been related to welldefined physical characteristics of the photocatalyst particles, namely their size (or size distribution), shape, mass concentration, and albedo of the material. As all these characteristics can be independently assessed and/ or controlled, the model should be regarded as being fully predictive. The model developed belongs to the class of the twoflux models and it has been shown that it underestimates the rate of photon absorption in photoreactors, so that it should be regarded as a conservative model for photoreactor design purposes. By exploring the TFM features, the following have been shown: (a) For L/λR > 2 the fraction of incoming photons reflected from the photoreactor has already practically reached its maximum value; this value is simply given by R/2 if R < 0.5, while for R > 0.5 it can be immediately obtained from a simple explicit equation. (b) The normalized transmitted energy, if plotted on semilog plots, will practically give rise to straight lines

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intercepting the ordinate axis at 1 for values of R smaller than 0.5, while the intercept will be significantly smaller than 1 for greater R values. (c) For L/λR > 4-5 the radiant fluxes and the LVREA can be computed, with very good approximation, by means of the equations pertaining to the case of semiinfinite slabs. These last have the advantage of being as simple as those of ZRM, a very helpful feature for engineering purposes. In the first part of this paper (Brucato and Rizzuti, 1997) another model had been developed and experimentally validated for the case of zero reflectance (ZRM), which had been shown to overestimate the photon absorption in the system. Therefore, the two models together constitute a modeling couple that gives rise to predictions that bound the real behavior of photoreactors, hence setting up confidence limits for the predicted performance. It has been shown that the simpler ZRM can be confidently applied with good approximation to all cases in which albedo is not too large, in particular to cases where R < 0.2. For greater values of R (up to R ) 0.5) a reasonable estimate can be obtained by averaging the results predicted by the two models. For even greater albedo values, use of the TFM can still be made, resulting in semiquantitative information on the way the main physical parameters (Dp, m, L) affect photoreactor performance. It also provides a conceptual framework that can help reasoning about, and therefore the understanding of the real behavior of photocatalyst suspensions. Acknowledgment This research work has been carried out under financial support of the STEP Research Programme of E.E.C. (Contract No. CT91-0133). Notation ap ) projected area of one catalyst particle (m2) Dp ) catalyst particle diameter (m) Dp av ) average particle diameter (m) g ) backward radiant flux (einstein s-1 m-2) go ) backward radiant flux at rear wall of photoreactor (einstein s-1 m-2) G ) forward radiant flux (einstein s-1 m-2) Go ) radiant flux at front wall (x ) 0) (einstein s-1 m-2) GL ) radiant flux at rear wall of photoreactor (x ) L) (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 mass concentration (kg m-3) np ) number of particles per unit volume (m-3) R ) catalyst albedo (no. reflected photons/no. interacted photons) 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) Φ(φ) ) phase function κ ) volumetric absorption coefficient (m-1) η ) dimensionless parameter defined in eq 11 φ ) angle between scattered photon and incident radiation λo ) extinction length for the case of R ) 0 (eq 13) (m) λR ) extinction length for the case of R > 0 (eqs 12 and 13) (m) Fp ) catalyst density (kg m-3) σ ) volumetric scattering coefficient (m-1) Ψ ) dimensionless parameter (eq 19)

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Received for review May 8, 1996 Revised manuscript received December 6, 1996 Accepted February 4, 1997X IE960260I

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