A Rational Approach to the Design of Photocatalytic Reactors
Giovanni Camera Roda* and Francesco Santarelli
Dipartimento di Ingegneria chimica, mineraria e delle tecnologie ambientali, UniVersita degli studi di Bologna,Viale Risorgimento 2, I-40136 Bologna, Italy
A model to assess the performances of an annular photocatalytic reactor has been developed by investigatingdifferent operational conditions through an analysis based on the proper dimensionless parameters (namely,the optical thickness, the Thiele modulus and two among the well-known Damkohler and Peclet numbers,which are properly redefined). Different dependences of the reaction kinetics on the local rate of radiantenergy absorption are also considered. Because the progress of the reaction is affected by the radiation field,all these parameters are dependent on the catalyst concentration and then indirectly on the catalyst load. Theconditions under which an optimal value for the catalyst concentration may exist are determined, thuscontributing to provide insight to one aspect that is quite controversial in the literature.
Introduction
It is expected that, soon, photocatalytic processes will bewidely used as an effective tool for the treatment of water thathas been polluted by traces of toxic and/or persistent chemicals.1
Therefore, a large amount of attention has been given to
studying the kinetics of the degradation of the organic pollutantsubstrate, as well as investigating the parameters that affect theperformance of the systems (lamp/reactor) used.
Results are undoubtedly interesting but suffer from a lack of generality,2,3 because they are presented in terms of dimensionalparameters, which are significant to the specific situationsinvestigated and, consequently, are valid under the samelimitations.
More-general criteria then are needed to assess the role thatthe operational variables have in determining and possiblyoptimizing the performance of a reactor.
A significant example of the previous statement is given bythe different conclusions presented by the authors who inves-tigated, in slurry systems, the effects of catalyst load on theconversion and possible existence of an optimal value for thisquantity.
According to a large number of authors,3-11 the rate of removal of the substrate increases monotonically with thecatalyst load and approaches an asymptotic value. The valueof the catalyst load at which the asymptotic behavior appearsis obviously dependent on the specific situation investigated;however, in any case, it is related to situations where a strongattenuation of the radiation field occurs within the reactor.
In other experiments and works,2,3,11-19 a maximum of therate of removal results at a specific value of the catalyst load.This behavior is generally justified by the significant attenuationof the radiation by the photocatalyst particles when theirconcentration is increased beyond a given value. This “shieldingeffect” might thwart the positive effect of an increase of theavailable catalyst sites, obtained by increasing the catalyst load.However, the maximum is strangely observed by the variousresearchers at catalyst concentrations that give very dissimilarattenuations of the radiation in the reactor, and this disagreementhas not yet been justified.
In a similar way, the dependence on the particular investigatedsituation limits the general validity of the analysis of the effects,
which are caused by (i) the flow rate of the stream to betreated2,20 and (ii) the order of the reaction, with respect to thelocal rate of radiant energy absorption.
The discrepancies that emerge in the conclusions of the citedinvestigations are not considered as a sign of inaccuracy but
are simply the outcome of attention to operational conditionswith a different relative weight of the relevant parameters.Therefore, the present contribution is an effort to investigate
the behavior of a photocatalytic reactor on the basis of thedimensionless parameters that are currently used in the transportphenomena and/or in the reaction engineering analysis to findanswers to some open questions.
Mathematical Model: Basic Equations
A photocatalytic reaction has been considered to occur withinan annular reactor with a fluorescent UV-A lamp placed on itsaxis. Because of the axial symmetry of the investigatedsituations, a two-dimensional problem results for both theradiation and the pollutant concentration fields.
The geometry of the system can be represented by thefollowing dimensionless variables:
where Rint is the internal radius of the annular region, R theexternal radius of the annular region, Rlamp the radius of theemitting lamp, and L the length of the irradiated annular reactor.
The investigation has been developed through the followingsteps of the analysis:
(a) radiative field and distribution of the rate of radiant energyabsorption within the reactor;
(b) conversion per pass when the reactor operates in acontinuous mode; and
(c) overall conversion when the reactor is inserted in a recycle/ batch system.
Radiative Field and Distribution of the Rate of Radiant
Energy Absorption. It is well-known that a peculiar featureof the photocatalytic processes is the essential role of the
* To whom correspondence should be addressed. E-mail address:[email protected].
radiation field as an inherent element of the process. This featureis common to all the photochemical processes and requires aspecific treatment that has a scaring effect on most of the peopleinterested in photochemical and photocatalytic reactions and isoften handled with simplified and questionable approaches.
A rigorous analysis of the radiative transfer can be developedstarting from the basic treatment presented in the textbooks onthis subject21,22 or from the comprehensive analysis of the roleof radiative transfer in photochemical processes.23-26
In the case of photocatalytic processes, the radiation field isdependent on the catalyst concentration after the geometry of the reactor and the location and the emission properties of thelamp are assigned. Even if the catalyst load is currently used toaccount for the radiation absorption, its intensive value, thecatalyst concentration, is definitely more appropriate for thispurpose, because it enters the constitutive equations for theoptical properties of the participating particles.
After the radiative transfer equation (RTE) has been solvedand the distribution of the radiation intensity has been obtained,it is possible to determine the local rate of radiant energyabsorption (e′′′) as
where ω represents any direction of the travelling radiation.
The relevant optical parameters that affect the radiativetransfer are
(1) the optical thickness, given as τ ) β R(1 - k ), where β )κ + σ is the extinction coefficient (here, κ is the absorptioncoefficient, and σ is the scattering coefficient),
(2) the single scattering albedo, ω0 ) σ / β, and
(3) the phase function p(ω f ω′) for elastic scattering fromthe ω-direction to the ω′-direction.
After the catalyst has been chosen, the albedo ω0 and the
phase function are assigned, because the absorption and scat-tering coefficients (and, then, the total extinction coefficient β)are dependent linearly on the catalyst concentration.27 Therefore,the effects on the photons distribution due to the catalyst loadcan be investigated more properly, only in terms of the opticalthickness, because this parameter accounts for the combinedeffect of the catalyst concentration (and then, indirectly, thecatalyst load) and of the dimensions of the reactor.28
Continuous Annular Photoreactor-Conversion per Pass.
Two flow conditionss fully developed laminar flow andturbulent flowshave been considered for the annular photore-actor when it operates in a continuous mode, assuming that thereaction rate is given by the kinetic equation
where e′′′ is the local rate of radiant energy absorption (here,e′′′ will be considered to be due to a monochromatic radiationor representative of the value averaged in a range of wave-lengths), R is the order of the reaction (with respect to e′′′),and C A is the substrate concentration. The assumption of a first-order reaction, with respect to the pollutant concentration, isconsistent with the low concentration values that are typical of most of these processes.
(a) Laminar Flow in the Reactor. When the Reynoldsnumber (which is defined as Re ) [2⟨V z⟩ R(1 - k )F]/ µ) is <2000,the flow in the reactor is laminar.
The following dimensionless variables have been defined:
where the characteristic concentration, C A1, is the concentrationat the inlet of the reactor;
where I 0 is the reference intensity of the radiation, which isdefined through the UV-A radiant power P emitted by the lampas I 0 ) P /(2π Rint L);
where D is the diffusivity of the organic pollutant substrate.Thus, the dimensionless mass balance equation can be written
as
where
Equation 2 has been solved for the following boundaryconditions:
Two dimensionless groups appear in the previous equations,namely, the Damkohler number,
(where V r is the volume of the reactor and V ˙ ) ⟨V z⟩π R2(1 - k 2)
C *A )C A
C A1
(e′′′)* )
e′′′
e′′′0 )
e′′′
κ × I 0
z* )z
R
r * )r
R
V* ) V ×R
D
1
r *
∂
∂r *(r *V*r C *A) +
∂
∂ z*(V* zC *A) )
1
r *
∂
∂r * (r *∂C *A
∂r * ) +
∂
∂ z* (∂C *A
∂ z* ) - φ2(e′′′*)R
C *A (2)
V* z ) 2(φ
2 L*
Da )
1 - (r *)2 + [1 - k
2
ln(1/ k )] ln(r *)
1 + k 2 - [1 - k
2
ln(1/ k )](3)
V*r ) 0 (3′)
C *A ) 1 (at z ) 0 and k < r * < 1)
∂C *A
∂ z*) 0 (at z ) L* and k < r * < 1)
∂C *A
∂r *) 0 (at r ) k and 0 < z* < L*)
∂C *A
∂r *) 0 (at r ) 1 and 0 < z* < L*)
Da ) K (e′′′)RV r
V ˙
e′′′ ) κ∫4π I ω dω
R ) K (e′′′)RC A (1)
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is the volumetric flow rate), and the Thiele modulus,
The choice of the two intervening groups is not unique, because,for instance, the Damkohler number Da, the Thiele modulus φ,and the mass Peclet number (defined as Pem ) Re(ν / D ) ) Re× Sc) are linked each other through the relationship Da × Pem
) φ22(1 - k ) L*.(b) Ideal Case of Plug-Flow Reactor. When the Re value
is high, the assumption of laminar flow does not hold any longerand a plug-flow reactor can be considered with a distributionof e′′′, as it is given by the solution of the RTE.
Indeed, at high values of Re, a negligible radial gradient of the substrate concentration and of the velocity can be assumed,because the mass boundary layer and the velocity boundary layerare both very thin.29 Also, the transport by diffusion in the
z-direction can be safely neglected, with respect to the convec-tive transport.
In this case, the mass balance equation for substrate A is
Assuming the kinetic equation as given in eq 1, the integrationof eq 4, with the inlet condition (at z ) 0) C A ) C A1, gives theoutlet concentration C A2 as
or, in dimensionless form,
It must be observed that the conversion, χP ) 1 - (C A2 / C A1), isa function of Da, k , L*, and the e′′′ distribution.A More-General Kinetic Equation. A more-general equa-
tion for the reaction rate can be written as
This equation accounts for a continuous change of the order of reaction,28 which is 1 at a low level of the intensity of radiation(c . e′′′) and, through a value of 0.5 at intermediate levels of the intensity, reaches the limiting value of 0 at a high level of intensity (c , e′′′). This occurrence has been experimentallyobserved and discussed by many researchers,9,28,30,31 so that,even if the underlying reasons are still under discussion, it can
be stated that eq 5 is able to represent at least the phenomeno-logical behavior of many real systems.
In this case, the mass balance equation in the reactor, in thehypothesis of no radial gradient of the substrate concentrationor of the velocity, becomes
Integrating this equation with the boundary condition that, at z) 0, C A ) C A1, the conversion turns out to be
where
Recycle/Batch System: Transient Behavior with the
Reactor in Pseudo-Steady State. The dynamic behavior of arecycle/batch system (see Figure 1) has been investigated,assuming that the reactor is operating in a pseudo-steady state.Under this assumption, because the reaction rate is first orderwith respect to the substrate concentration, the conversion perpass in the reactor is independent of the inlet concentration and,consequently, does not change during the process.
The condition of pseudo-steady state for the reactor can besafely assumed if the residence time in the reactor ( t r) is muchshorter than the characteristic time of disappearance of the
substrate from the system (t d). Because of the fact that thesetwo characteristic times can be estimated as
and
where V t is the volume of the perfectly mixed tank and C A0 isthe initial concentration of the substrate in the system, the reactorworks in pseudo-steady state if
φ ) R K (e′′′0 )R
D
V ˙dC A
d z
)∫kR
RR 2π r dr (4)
C A2) C A1
exp[-K
V ˙ ∫V r(e′′′)R dV ]
C A2
C A1
) exp[-Da
π (1 - k 2) L*∫
V *r (e′′′*)R dV *]
R ) K (e′′′
e′′′ + c)C A (5)
V ˙dC A
d z) -KC A∫kR
R e′′′
e′′′ + c2π r dr (6)
χP ) 1 - exp[-Da
π (1 - k 2) L*∫
V *r
(e′′′)*
(e′′′)* + AdV *]
Figure 1. Schematic setup of the batch/recycle system.
(e′′′)* )e′′′
e′′′0
A )c
e′′′0
Da )KV
rV ˙
t r )V r
V ˙
t d )C A0
(V r + V t)
rate of disappearance of A from the system
)C A0
(V r + V t)
K (e′′′0 )RC A0
V r
)V r + V t
K (e′′′0 )RV r
t r
t d)
K (e′′′0 )RV r
V ˙×
V r
V r + V t
) Da( 1
1 + V t / V r) , 1
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i.e., it is required that Da , 1 and/or V r / V t , 1. Note that theselatter inequalities are very often satisfied in real cases.
The mass balance of A in the stirred tank is
In pseudo-steady state and in the hypothesis that the volumeinside the tubes is negligible, with respect to V t, the concentration
C A2 is given, at any time, by C A2 ) C A1 × (1 - χp), where χpis the conversion per pass in the reactor.
Introducing the dimensionless time, t *) t × R2 / ν, the massbalance equation becomes
where K d is the dimensionless constant of the disappearance of A in the system (K d ) { Re /[2(1 - k ) L*]}(V r / V t) χp). Therefore,only one additional parameter, the ratio V r / V t, must be consideredto characterize the transient behavior.
The result of the integration of the mass balance equationwith the initial condition that C A1 ) C A0 at t ) 0 is
Discussion of the Results
The values of the geometrical dimensionless variables havebeen assumed, with reference to an experimental annular reactoravailable in the authors’ laboratory, to be as follows:
For the optical properties of the photocatalyst particles, referencehas been made to Degussa P25 titanium dioxide, assumingisotropic scattering and averaged values in the wavelength rangeof 300-400 nm for the absorption and the scattering coef-ficients, based on the data measured by Cabrera et al.27
The RTE (radiative transfer equation), with the properhypothesis and boundary conditions, which are illustrated bySgalari et al.,25 has been solved by the finite volume method.32,33
The obtained values of e′′′ are then utilized in the source
term of eq 2, which has been discretized by the finite volumemethod34 and numerically solved.
The role of τ in affecting the radiation field can be clearlyunderstood by a combined scrutiny of Figures 2 and 3, as thevalue of this parameter increases when the catalyst load and/orthe annular gap ( R × (1 - k )) of the reactor increase.
When τ increases, the fraction of the emitted power that isabsorbed within the reactor increases, but, at the same time, anincreasing nonuniformity results for the distribution of e′′′ andthen, even if the absorbed power increases, a less-effectiveexploitation of the reaction volume occurs.
Therefore, it can be expected that the resulting effect on theconversion is dependent on which of the two competitiveoccurrences is the prevailing one. Because the distribution of
the absorbed photons affects the local rate of disappearance of
the material reactant, it turns out that, when a large nonunifor-mity occurs in the distribution of e′′′ (i.e., for large values of
τ ), the diffusion of the reacting species may have a significant
role.
After the effect of the optical thickness on the radiation
distribution has been evaluated, the influence of τ on the
conversion per pass has been investigated for two values of R(R ) 0.5 and 1) and for the two flow regimes considered.
In Figure 4, the conversion per pass is given, relative to the
optical thickness τ , at various values of the Reynolds number
( Re) for R ) 0.5 and 1.
Note that, to make the role of the flow regime more evident,
the more familiar Re has been assumed to be one of the two
independent dimensionless groups upon which the solution of
the problem is dependent. This choice is consistent with the
possibility outlined in the previous section and considering that
Re and Pem differ by a multiplying constant, as the Schmidt
number (Sc) is constant for a given fluid.
The case of Re ) 10000 has been considered to be
representative of turbulent flow conditions.
Because τ accounts for the catalyst concentration that, through
e′′′0, affects also Da and φ, it turns out that, if the geometrical
dimension is kept constant, a different value is obtained at any
τ for Da and for φ. The curves in the figures have been obtained
V tdC A1
dt ) V ˙ (C A2 - C A1) (7)
dC A1
dt *) -K dC A1
(8)
C A1
C A0
) exp(-K dt *)
R ) 20.5 mm
R*lamp )8
20.5
k )12
20.5
L* )265
20.5
Figure 2. Radial distribution of e′′′* at z* ) L*/2, at different values of τ .
Figure 3. Normalized absorbed radiant power, as a function of τ .
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assuming φ ) 50 at τ ) 10, based on realistic values for thekinetic constant K and for the substrate diffusivity D .
The results show the following:(1) Conversion obviously decreases when Re increases, if the
other conditions remain unchanged, because of the reduced meanresidence time. Mean residence time is actually the mostsignificant fluid dynamics parameter that affects the performanceof the reactor, whereas the residence time distribution (RTD)has a marginal role on the conversion, as will be shown later.
(2) Different situations occur for different values of R: forR ) 0.5, a maximum always results for the conversion when τ increases. Even if this seems to be evident only for Re ) 250in the R ) 0.5 case, this conclusion is, on the contrary, quite
general and it would be clearly apparent if the conversion isnormalized with respect to its maximum value.
Some relevant radial concentration profiles at an axial positionclose to the exit of the reactor are examined in Figure 5. Thedisappearance of the substrate is higher in the positions wherethe residence time is higher; that is, the axial velocity is low,and e′′′ is larger (close to the inner wall). As a consequence,because the diffusion is rather ineffective in transporting thesubstrate in the radial direction, the profiles do not seem to beuniform. This effect is particularly evident when the extent of reaction is high; that is, when τ is high or Re is low, as it isapparent in Figure 5.
A utilization of the reactor with a single pass in a continuousprocess is often impracticable, because of the very low conver-
sion per pass. For this reason, in many applications, the outletstream is recycled to the reactor so that a reasonable conversioncan be obtained after a sufficient time of “batch” operation of the system.35 Very often, a tank is added to the system, as shownin Figure 1. The function of the tank can be (i) to control theconditions of the suspension during the operation by agitating,aerating, and thermostating it, and/or (ii) to increase the volumeof the aqueous solution to be processed which, otherwise, shouldbe limited to the usually small capacity of the reactor and of the tubes.
The transient behavior has been simulated by the modelpresented in the previous section to obtain the variation withtime of the concentration of the substrate in the tank with aratio V r / V t ) 0.05. The results obtained in the simulations showthat, in the case of no radial gradient of concentration and of
velocity within the reactor, the rate of disappearance of thesubstrate in the system is almost independent of the recycle flowrate. Therefore, the results obtained for Re ) 10 000 are actuallyrepresentative of whatever the flow rate in turbulent flow. Theyare plotted in Figure 6 for different values of τ , which, again,represent different values of the concentration of the photo-catalyst.
When R ) 0.5, the fastest decay is observed at a certain valueof τ (τ = 5), whereas when R ) 1, the decay is faster withlarger τ . The results can be examined also through a scrutinyof the constant of disappearance K d, which has been previously
Figure 4. Conversion per pass, as a function of τ at different values of Re: (a) for R ) 0.5 and (b) for R ) 1.
Figure 5. Radial concentration profiles at z* ) L*: (a) at different values of τ for Re ) 250 and R ) 0.5 and (b) at different values of Re for τ ) 10 andR ) 1.
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introduced and characterizes the exponential decay of thesubstrate in the system.
In Figure 7, K d is plotted against τ for different values of Re
in the reactor, i.e., for different recycle flow rates. The followingobservations can be observed:
(1) For R ) 0.5, a relative maximum of K d is always presentand then an optimal value of the optical thickness results, i.e.,an optimal value of the concentration of the photocatalyst.
(2) For R ) 1, a relative maximum of K d is present only forlaminar flow, when the radial profile of the substrate concentra-
tion is not flat, because of limitations in the mass transport of the substrate in the radial direction. Otherwise, K d is monotoni-cally increasing with τ toward an asymptotic value.
(3) At a given τ , the variation of K d and, hence, of the rateof disappearance of the substrate with Re is more remarkableat the highest values of τ and R; however, in any case, the effectof the flow rate is not very important. As a matter of fact, evenif the substrate concentration profiles can greatly diverge atdifferent Re (see, e.g., Figure 5b), this variation, nonetheless,is mainly confined in a zone where the convective transport inthe z-direction is low, because the velocities here are low and,thus, the contribution to the conversion is limited. For the sakeof comparison in Figure 5b also, the concentration profile at
Re ) 250 is plotted for the hypothetical case of no radial
gradient of the velocity and of the concentration. The molarrate of the substrate at the exit is proportional to the integralfrom the inner radius to the outer radius of the local concentra-tion weighted by the local velocity and the local radius. Thefinal result of this “weighted” integration is that the differencesare smoothed and, hence, the conversion also is not greatlyaffected by the transport limitations of the reagent in a radialdirection. In other words, diffusion can significantly affect theconcentration profiles inside the reactor, but it appears to beunable to severely limit the rate of disappearance of the
substrate. Besides, the limitation of the diffusion on the rate of disappearance is assessable only at rather high values of thecatalyst concentration, which represent cases of less-practicalimportance, because it is useless to increase the catalyst loadadditionally when no appreciable further enhancement can beobtained.
The analysis of the effect of the order of reaction can beextended by considering the more-general kinetic eq 5 previ-ously presented. It has been used to analyze the effect of achange of the concentration of the photocatalyst and of thereference intensity ( I 0) on the observed rate of disappearanceof the substrate, which is soundly reproduced by the value of K d in the batch apparatus. Only the case of turbulent flow inthe reactor has been considered (without any radial gradient of
Figure 6. Tank bulk concentration, as a function of time, in the case of turbulent flow at different values of τ : (a) for R ) 0.5 and (b) for R ) 1.
Figure 7. Dimensionless kinetic constant of the disappearance of the substrate, as a function of τ , at different values of Re: (a) for R ) 0.5 and (b) for R) 1.
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concentration and velocity). An increase of the concentrationof the catalyst implies an increase of the absorption coefficientκ, and, as a consequence, as previously discussed, it can be
reproduced by a variation of the optical thickness τ . Differentvalues of I0 can be represented by the corresponding values of the parameter A ) c / e′′′0 ) c /(κ I 0) at a given optical thickness.
The results obtained are plotted in Figure 8 versus the opticalthickness for different values of the parameter A evaluated at τ ) 10.
At very low values of I 0 (high values of A), the kinetics isobviously slow and K d is very low. Moreover, in this case, onlya first-order relationship is effective throughout the entirereactor, because of the low level of e′′′ with respect to c at anyradial position. The consequence is that no relative maximumis found by varying the optical thickness, that is, by varyingthe concentration of the substrate. These results are qualitativelysimilar to those previously obtained for R ) 1.
In contrast, at low values of A, the order of the reaction maychange inside the reactor with the radial position, in particular,if the optical thickness is high. In fact, in this latter case, bytaking into consideration also the radial distribution of e′′′reported in Figure 3, at any radial position a different order of reaction is actually effective. In particular, at a radial positionclose to the emitting lamp, e′′′ is much higher than c and theorder of the reaction is zero, whereas, at a radial position farfrom the lamp, e′′′ is much lower than c and the order of thereaction is 1. At intermediate radial positions, the entire rangeof order of reaction, from 0 to 1, is experienced.
The final result of the complex interaction between theabsorption of the radiant energy and the kinetics of the reactionis that, at high values of the reference intensity I 0, a relative
maximum of K d appears, as is evident in Figure 8. Thismaximum is more pronounced with lower A (or higher I 0) valuesand the corresponding optimal value of the catalyst concentrationdiminishes by increasing I 0.
Conclusions
It has been demonstrated that the rate of disappearance of anorganic pollutant substrate in a photocatalytic slurry is dependenton a reduced number of dimensionless parameters. Specifically,the geometric ratios of the system, the optical thickness, theDamkohler number (or, alternatively, the Reynolds number( Re)), and the Thiele modulus are the dimensionless groups thatdefine the performances of the system, together with an
additional parameter A, which takes into account the kineticequation of the rate of reaction.
When the dimensions of the reactor are fixed, the opticalthickness becomes dependent linearly on the catalyst concentra-tion. With the present dimensionless approach, it has been shownthat the rate of disappearance of the substrate from the systemexhibits a maximum at an optimal value of the catalystconcentration only at low Re values or at low A values.Otherwise, the rate of disappearance is continuously increasing
with the catalyst concentration toward an asymptotic value.In other words, it has been demonstrated that the occurrence
of the maximum and also its possible “position” versus thecatalyst concentration are affected by the following parameters:
(1) The intensity of the entering radiation, because, ultimately,the intensity may affect the order of the reaction, with respectto the local rate of radiant energy absorption. In particular, thehigher the intensity of the impinging radiation, the higher thepeak of the rate of removal of the substrate and the lower thecatalyst load that gives this maximum.
(2) The flow rate. However, the effect of the flow rate isminor.
The previous observations demonstrate that the occurrenceof a maximum for the rate of substrate disappearance cannot
be explained only based on the “shielding effect”, but it is duealso to other concurrent phenomena, namely, the diffusion of the substrate and the influence that the radiation power enteringthe reactor has on the kinetics of the reaction. As a consequence,the optimal value of the catalyst concentration is dependent onthese phenomena.
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