LVRPA

The relationship between the local reaction rate and the LVRPA can be obtained from models of the reaction mechanism. By considering the steps proposed by Turchi and Ollis (1990), and taking into accoimt the simplifications proposed by Bandala et al. (2004), it is obtained... 

The above equation gives the local reaction rate as a product of two functions, one dependent on the concentration of fhe pollutant and the intermediates only, and a second one dependent only on the LVRPA. Note that all the concentrations are location dependent as well as Cl- To obtain the evolution of the average concentration of fhe pollutant it is necessary to average Equation (8) over the suspension volume Vj... 

The determination of the distribution of the LVRPA requires the use of some type of radiative transfer model. In the case of transparent pollutants, it can be considered that Cl depends on Ti02 concentration (Qatai) only, and not on the concentration of the pollutant, since it is the former component which absorbs and scatters radiation. This allows imcoupling the radiation problem from the degradation kinetics when Equation (13) is solved that is, one can first evaluate and then, independently of the value of the pollutant concentration, integrate F2(Cl) over the reactor volume. Once this quantity has been calculated, its numerical value is taken as a constant in Equation (13), which can now be solved to obtain the evolution of Cp av... 

These equations mean that a linear dependence of the reaction rate with light intensity (LVRPA) is observed when intensities are small, while square root dependence is observed when intensity is high. This latter dependence occurs for high intensities because the recombination of electron-hole pairs starts to limit the efficient use of the available photons (Alfano et al., 1997). The intensity at which the crossover between these two types of behavior occurs depends on the value of the lumped kinetic parameter 7, which in turn depends on the specific reaction under consideration. 

For small values of 7, when the kinetics is linear with respect to the LVRPA, this integral increases asymptotically with catalyst concentration (Arancibia-Bulnes and Cuevas, 2004). Flowever, when square root dependence dominates, this factor reaches a maximum and then starts to decrease with increasing catalyst concentration. This means that too much catalyst has a negative optical effect at high radiation intensity leading to smaller degradation rates. This behavior is illustrated in Figure 8. 

LVRPA local volumetric rate of photon absorption... 

It is rather atypical that a photochemical reaction will proceed in a single molecular pathway. Thus, several elementary steps are involved. Normally, the majority of them are dark (thermal) reactions while, ordinarily, one activation step is produced by radiation absorption by a reactant molecule or a catalyst. From the kinetics point of view, dark reactions do not require a different methodological approach than conventional thermal or thermal-catalytic reactions. Conversely, the activation step constitutes the main distinctive aspect between thermal and radiation activated reactions. The rate of the radiation activated step is proportional to the absorbed, useful energy through a property that has been defined as the local volumetric rate of photon absorption, LVRPA (Cassano et ak, 1995 Irazoqui et al., 1976) or the local superficial rate of photon absorption, LSRPA (Imoberdorf et al., 2005). The LVRPA represents the amount of photons that are absorbed per unit time and unit reaction volume and the LSRPA the amount of photons that are absorbed per unit time and unit reaction surface. The LVRPA is a property that must be used when radiation absorption strictly occurs in a well-defined three-dimensional (volumetrical) space. On the other hand, to... 

The reaction rate requires the evaluation of the LVRPA. In the absence of emission and scattering, the transport of photons in the... 

In Equation (36) the subscript P was included to indicate that in calculating the LVRPA from the value of G x, t) only the absorption coefficient of the reactant radiation absorbing species must be included. When Equation (36) is averaged over the reactor volume, it renders the required expression for Equation (32). This direct substitution is possible only because the reactor is assumed to be well mixed and concentrations are uniform, taking on a... 

Figure 13a, b shows two t)q5ical outcomes where the solid lines are the theoretical simulations from the simplified model. These parameters can be now applied to the large-scale reactor and, as before, a new value of the LVRPA will be necessary according to its particular configuration. 

Finally, af any poinf inside fhe annular reaction space, the LVRPA is... 

In Equations (51)-(54), it must be noted that (i) e (x) = e [(x), Can] to recall that the LVRPA is a strong fimction of the catalyst loading and (ii) there are eight lumped kinetic parameters that depend on intrinsic properties of the system according to... 

Equations (51)-(54) indicate the need for calculating and The evaluation of the LVRPA inside the reactor was achieved by solving the RTE for the heterogeneous system. The radiation model considers that... 

The LVRPA for polychromatic radiation is calculated from the values of the monochromatic radiation intensity as... 

LVRPA profiles for different catalyst concentrations and different reactor... 

Cm = 0.05 X 10 g cm, the reaction rate is proportional to the square root of the LVRPA throughout the reaction space. Consequently, in a photoca-talytic slurry reactor, where low, intermediate, and high photon absorption rates can coexist, the complete reaction rate equation must be used. 

The existence of the LVRPA in the reaction rate expression (Equation 75) makes necessary the solution of the RTE in the pilot scale reactor. As explained before, the lamps length is significantly larger than the reactor width Yr, thus imiformity of radiation is considered along the y direction. Therefore, a two-dimensional (x, z) model for the spatial variations of the radiation field was adopted. The angular distribution of radiation was modeled with the spherical coordinates 6, (j>). The RTE for a two-dimensional, two-directional model is (Brandi et al., 1996, 1999),... 

The discrete ordinate method (Duderstadt and Martin, 1979) was employed to solve the RTE (Equations 82 and 83). Afterward, the LVRPA was obtained according to... 

Where [0i, 62] and [< i, ( 2] are the integration limits that define the space from which radiation arrives at the point of incidence. For each point of incidence, in practice, these limits are defined by the extension of the lamp (its diameter and its length). Thus, to evaluate the LVRPA we must know the spectral specific intensity at each point inside the reactor. Its value can be obtained from the photon transport equation (equation 6.23). 

This ordinary differential equation must be solved with the initial condition indicated in equation 6.63. Note that due to the required averaging procedure, the reaction order with respect to the LVRPA (n) has a rather complex relationship with respect to the time rate of change of concentrations. 

The boundary condition was obtained from the 3D with voluminal and isotropic emission model (equation 6.51). The solution of equation 6.23 provides values of the radiation intensity as a function of position (r, z) and direction (, ). Once is known, the incident radiation and the LVRPA can be obtained from equations 6.24 and 6.25. Since monochromatic radiation is employed, no integration over wavelength is needed. The final equation for calculating the LVRPA is... 

The integration limits for 6 and (f> for the case of the annular reactor are described by equations 6.52, 6.53, and 6.54. It must be noticed that the exponential term (attenuation) uses the reacting medium total absorption comment while only the reactant absorption coefficient intervenes, with a linear effect, in the value of the LVRPA. Hence i stands... 

In this reaction, the concentration of the absorbing species remains constant for conversions below 20% (sensitized reaction). To validate the radiation model, results obtained from equation 6.74 must be compared with experiments. From equations 6.72 and 6.73 and after integration, the experimental value of the LVRPA is... 

Inserting the LVRPA into equation 6.78 and substituting the result into equation 6.76 we obtain... 

Photon absorption rate by a material particle of the suspension. At this point we would like to know the LVRPA by the solid and to be able to isolate this value even if the liquid would also absorb radiation. To do this we need to model absorption by a material particle of the suspension. In the continuum mechanics sense, a material point in space is a volume for which every property can be well defined by a single value. For a catalytic suspension, it will be made of the liquid and the solid phases. Let us consider a small volume V of the suspension space representing this material particle. This volume is located at a point in space x (Figure 6.11). Any point inside V can be defined in terms of a local reference frame f. 

We must now relate the LVRPA per particle, (x-l-, f), to the LVRPA by the suspension volume (liquid -I- solid), e (x, t). The absorbed energy per unit wavelength interval, unit time, and unit volume of the suspension (solid plus liquid) is by definition of an average value over the total volume ... 

Calculating procedure for the LVRPA. In order to apply equation 6.95 we need to solve the RTE (equation 6.32) for this particular reactor set-up. As shown by Alfano etal. (1995) and Cabrera etal. (1994) the radiation field of this reactor can be modeled with a ID, one-directional radiation model and rather simple boundary conditions (Figure 6.10). Hence, with azimuthal symmetry derived from the diffuse emission at x = 0 ... 

Equation 6.96 with boundary conditions 6.97 and 6.98 can be solved with the DOM (Duderstadt and Martin, 1979). Solution in terms of intensities can be immediately used to calculate local values of the LVRPA. Optical properties can be assumed constant (stable catalyst) and, consequently, for a transparent organic compound the and (T), values are only a function of position at most. The numerical result gives monochromatic... 

The LVRPA or the photon absorption rate as a function of position results in ... 

The absorption coefficient has been assumed independent of position (uniform catalyst concentration) and time (stable catalyst). The reactor volume average of the LVRPA for the ID model in the Cartesian coordinate x is... 

Three different but connected problems must be studied (i) the reaction kinetics model (ii) the development of the rate of electron-hole generation in a material particle of the solid suspension and (iii) the model for characterizing the radiation field to evaluate the local volumetric rate of photon absorption (LVRPA). Point (iii) has been already described in section 6.6.1 for quantum yield determinations. In the first part of this section, we will concentrate on problems (i) and (ii). 

In equation 6.118, the monochromatic LVRPA [e (x, t)] is the result of the average absorption rate calculated over all the catalytic surface (area Ag corresponding to Vg) existing in the volume V. The LVRPA must be obtained from the solution of RTF applied to each particular reactor. 

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