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Diffusion limitations in stagnant photocatalytic reactors
Chemical Engineering Journal 247 (2014) 314–319
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Chemical Engineering Journal journal homepage: www.elsevier.com/locate/cej
Diffusion limitations in stagnant photocatalytic reactors Mahsa Motegh 1, Jiajun Cen 1, Peter W. Appel, J. Ruud van Ommen, Michiel T. Kreutzer ⇑ Department of Chemical Engineering, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands
h i g h l i g h t s Analyzes of uneven absorption of light in photoreactors vs. counteracting diffusion. Describes the extremes of optically thick reactors and optically thin reactors. Provides a criterion for maximum allowable reaction rate for kinetic experiments.
a r t i c l e
i n f o
Article history: Received 1 November 2013 Received in revised form 13 February 2014 Accepted 19 February 2014 Available online 7 March 2014 Keywords: Photocatalysis Slurry reactors Diffusion limitation Photon absorption rate Mathematical modeling
a b s t r a c t This paper provides a simple criterion to determine when the performance of an unmixed photocatalytic slurry reactor becomes limited by diffusion. We use a 1D description of the reactor and the two-flux intensity model to describe the concentration profile in unmixed photoreactors. We show that the effect of diffusion limitation in optically thick photoreactors is negligible when the Dahmköhler number based on reactor length is smaller than 0:1 sc, where s is the optical thickness and c is the exponent that describes how the reaction rate varies with light intensity. For optically thin reactors, in contrast, we find that the maximum Dahmköhler number scales with the inverse of sc. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction This paper explores the effect of mass-transport limitations on photocatalytic reactors. Invariably, when mass transport is not fast enough to keep up with catalysis, the overall reaction rates changes. Classical examples include the effect of mass transport outside a catalytic particle [1], mass transport inside a catalyst particle [2] and axial dispersion in nearly-plug-flow reactors [3]. For all of these examples, the most useful analysis has been to define a criterion that teaches when change of the rate due to limitations exceeds a given threshold, typically 5%. Such an analysis captures the result of approximate or exact solutions of the governing component balances in a single criterion that can be used by experimentalists. For photocatalytic reactors, such criteria are largely absent. The situation in photocatalysis is more complex: in addition to a nonuniform concentration field c, the optical field variable of relevance, i.e. the rate of photon absorption ea by the particles, is also not constant throughout the reactor. This local volumetric rate of ⇑ Corresponding author. 1
E-mail address: [email protected] (M.T. Kreutzer). Both authors contributed equally.
http://dx.doi.org/10.1016/j.cej.2014.02.075 1385-8947/Ó 2014 Elsevier B.V. All rights reserved.
photon absorption ea appears in the kinetic expression r ¼ f ðc; ea Þ. Gradients in ea lead to gradients in r, which in turn lead to concentration gradients, even if the catalyst is homogeneously dispersed and particle-level gradients can be ignored. Of course, vigorous stirring can eliminate such concentration gradients, but we find many examples [e.g. 4,5] of unstirred catalytic performance tests that lack forced convection (e.g., stirring) or reported natural convection (e.g., due to heating by incoming light or electron/hole recombination). Herein, we analyze the simplest case of mass transport limitations in such stagnant photoreactors. To keep the problem tractable, we consider a reactor volume between parallel transparent plates, with light of intensity I0 entering perpendicularly onto one side, and we use a two-flux model for the radiation transfer, as discussed below. This renders the problem one-dimensional. After a brief formulation of the problem, we define our five-percent criterion for the onset of mass transport limitations. We then explore limiting cases of optically very thin and optically very thick reactors. This gives analytical expressions for the asymptotes of the criterion, which have the most relevance in catalyst performance tests, and we use numerical calculations for the intermediate regime that is more important in design of efficient photoreactors. We mention here that there are other aspects of
M. Motegh et al. / Chemical Engineering Journal 247 (2014) 314–319
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unmixed catalyst testing, such as aggregation and sedimentation of catalyst particles, concurrent possible diffusion limitation of oxygen from the air, and the likely natural convection from heating on the illuminated side, that are beyond the scope of this paper.
pffiffiffiffiffi r / ea when electron–hole pairs are generated much faster than they can migrate to the surface. Here, we shall use a power-law a expression r ¼ kc eca , where the kinetic constant k can be any value and the kinetic constants 0 < a 6 1 and 0:5 < c < 1, typically.
2. Problem formulation
2.3. Governing component balance
2.1. Local rate of photon absorption
Without convection, the transient component balance for concentration cðx; tÞ only has a diffusion term and a reaction term:
We first describe the local rate of photon absorption inside the slurry. The simplest analysis just uses the Lambert–Beer equation that ignores all scattering, resulting in an exponential decay of the light intensity I inside the reactor, IðxÞ ¼ I0 expðbxÞ. The local rate of photon absorption is then given by
@c @2c c ¼ D 2 kea ca ; @t @x
ea ¼ dI=dx ¼ bI0 expðbxÞ:
ð1Þ
where b is the extinction coefficient and x is the spatial coordinate that varies from x ¼ 0 where the light enters to x ¼ L on the other end. A useful parameter is the optical thickness s ¼ bL. A reactor is thick (opaque) if s 1, indicating that very little light penetrates through the reactor. Conversely, a reactor is optically thin if s 1. Photocatalytic particles not only absorb light but also scatter it. Scattering occurs in all directions, and the full calculation of the radiation transfer equation is difficult. The two-flux model [6] simplifies the complex analysis of radiation transfer by assuming that light only travels in one direction. Light can be absorbed in a differential slice of thickness dx, or it can be scattered back, or it can pass through without changing path. The last option occurs because of a forward-scattering event or because the photon simply does not interact with any particle. Of course, backscattered light can be scattered again. For unidirectional light that enters the slurry at x ¼ 0, this results in
bx
ea ¼ bI0 a1 a2 e þ a3 e
bx
ð2Þ
where and a1 ; a2 ; a3 are dimensionless coefficients. The term a2 expðbxÞ in Eq. (2) complicates the analysis, but it turns out that for thick reactors, a2 is vanishingly small, such that we can write a pseudo-Lambert–Beer expression that consists only of an exponential decay term. Motegh et al. [7] showed that for optically thick photoreactors,
ea ¼ bI0 ½1 qðxÞebx
ð3Þ 1 2 1=2
where qðxÞ ¼ x½1 þ ð1 x Þ is the fraction of lost photons via backscattering out of the reactor. It depends on the scattering albedo x ¼ r=b, where r is a scattering coefficient. Limiting values of x are x ¼ 0 when particles do not scatter but only absorb and x ¼ 1 when particles only scatter but do not absorb light. 2.2. Kinetics To use the local rate of photon absorption in a reactor model, we need a kinetic expression that relates the absorption of photons by particles to chemical conversion at the particle surface. Many such expressions have been proposed. Based on microkinetics, one can derive expressions for the reaction rate per unit slurry volume r 1=2 of the form r ¼ k1 f ðcÞ½1 þ ð1 þ k2 ea Þ , where k1 and k2 are kinetic constants and f ðcÞ 2 ½0; 1 describes the surface saturation as a function of concentration c [8]. Most relevant photocatalytic processes are pollutant removals at low concentrations, for which surface saturation does not occur, such that the reaction is observed to be first order. At higher concentration, the observed reaction order can be smaller. The term in the square root describes the effect of electron–hole recombination. When ea is small, electron– hole pairs migrate faster to the surface than they are generated, such that recombination is minimal and r / ea . Conversely,
ð4Þ
with initial condition cðx; 0Þ ¼ c0 and no-flux boundary conditions @c=@xð0; tÞ ¼ @c=@xðL; tÞ ¼ 0. Note that r and ea are defined per unit volume of the slurry mixture, such that the catalyst concentration is ‘‘hidden’’ in k. We are interested in the concentration profile in the entire reactor, so we scale length as X ¼ x=L. The concentration falls from the initial value c0 to zero, so we have for dimensionless concentration C ¼ c=c0 . The characteristic time iis either the diffusion h 1 a1 time L2 =D or the reaction time kc0 ðbI0 Þc . The ratio of these characteristic times is the Dahmköhler number a1
Da ¼
kc0 ðbI0 Þc L2 : D
ð5Þ
The significance of Da is shown in Fig. 1: small values, in Fig. 1(a) and (b), indicate that gradients are small, whereas for Da 1 in Fig. 1(c) and (d), gradients in concentration are significant. We scale 1 time with the diffusion time, i.e. T ¼ ðL2 =DÞ t, and obtain
@C @ 2 C ¼ DaEca C a : @T @X 2
ð6Þ
where Ea ¼ ea =bI0 is the dimensionless rate of photon absorption. The scaled initial condition and boundary conditions are CðX; 0Þ ¼ 1; @C=@Xð0; TÞ ¼ @C=@Xð1; TÞ ¼ 0. 3. Criterion for mass transfer limitations A good instantaneous measure for diffusion limitations is the difference in concentration between the illuminated end and the dark end of the reactor. We define a time-averaged version over the entire conversion as
R1
¼
0
½Cð1; TÞ Cð0; TÞdT R1 Cð1; TÞdT 0
ð7Þ
and define as a criterion for the absence of mass transport limitations < 0:05. As discussed above, we limit ourselves to 0 < a 6 1. As is generally the case, for first-order reactions the concentration asymptotically approaches zero, whereas for fractionalorder reactions ð0 < a < 1Þ the concentration becomes zero in finite time. From numerical solutions of Eq. (6) where non-negativity of concentration was ensured, we found for first-order reactions that we could truncate the integration at T 10, with negligible change in upon continued integration. For fractional-order reactions, reaches a finite value when the concentration becomes zero everywhere, which happens in finite time. In other words, never diverges to infinitely for the range of a that we are interested in and gives a good time-averaged measure for the extent of diffusion limitations. The criterion ¼ 0:05 will depend on the Dahmköhler number Da, the optical thickness s, the scattering albedo x and the reaction orders a and c. The problem is well tractable for the first-order reactions without scattering, i.e. the case a ¼ 1; x ¼ 0, which we will describe in detail. This will reveal the important features of the boundary, which we write as ðDa; scÞ ¼ 0:05, because it will turn out sc always appear together as a group. Subsequently, we
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Increase in
X=τ -1
(a) 1.0 C
0.5
0.1
0.1
0.5
0.5
rxn region
rxn region 0.75
0.0
0.75
Ea
dark region
(d)
(c) 1.0
0.1
0.1
0.5
Ea
C
Increase in Da
Ea
-1 (b) X=τ
0.5
0.5
0.75
rxn region
0.75
rxn region
0.0 0
1
Ea dark region
0
1
X
X
Fig. 1. (a–d) The dimensionless concentration profile as a function of depth X when the average conversion is set to 10%, 50%, and 75% (solid lines) and the profile of the rate of photon absorption (dashed lines). (a and b) Low Dahmköhler number ðDa ¼ 0:1Þ for (a) optically moderately thin ðs ¼ 1Þ and (b) optically thick ðs ¼ 10Þ case. (c and d) High Dahmköhler number ðDa ¼ 1000Þ for (c) a optically moderately thin ðs ¼ 1Þ and (d) optically thick ðs ¼ 10Þ case.
shall see that using Eq. (3) permits the use of the zero-scattering result for thick reactors with scattering, such that only for sc 1 we need to use numerical calculations to describe situations where x > 0, i.e., situations with scattering in thin reactors. Finally, we will discuss numerical solutions for the situation a < 1.
CðX; TÞ ¼
4. Asymptotic behavior for thin reactors
ð8Þ
which is readily solved by separation of variables, resulting in CðX; TÞ ¼ exp½Dað1 scXÞT for first-order reactions and CðX; TÞ ¼ ½1 DaTð1 scXÞð1 aÞ1=ð1aÞ for fractional-order reactions. Substituting either of these into Eq. (7) and integrating until (for a < 1) the concentration becomes zero reveals that, for any reaction order 0 < a 1, irrespective of the value of the Dahmköhler number,
¼ sc:
1 X Y n exp k2n þ Dað1 scÞ T Un ðXÞ;
ð10Þ
n¼0
In thin reactors, such as shown in Fig. 1(a) and (c), we have nonzero photon absorption throughout the reactor. The worst-case scenario, as far as diffusion limitations go, is that diffusion is not able at all to smooth out the gradients caused by non-homogeneous reaction. This occurs at high Da, such that the ð@ 2 C=@X 2 Þterm is negligible with respect to the reaction term. Thus, a highDa situation can be approximately analyzed by dropping the diffusion term, such that the concentration is simply given by @C=@T ¼ DaEca C a . For non-scattering photon-absorption (Eq. (1)) in thin reactors, such that sX 1, this simplifies to
@C=@T ¼ Dað1 scXÞC a ;
reactions, a solution of the resulting balance can be expressed in terms of Airy functions. Using the finite Fourier transform method [9, chap. 5], a solution, in the form of an infinite sum using orthonormal basis functions Un ðXÞ, is readily found to be
ð9Þ
Thus, for thin reactors, the asymptotic behavior of the criterion for mass transfer limitations is that as sc goes to 0.05, Da may go to infinity. For sc < 0:05, the difference in the rate of photon absorption is so small that significant gradients in concentration across the reactor can never develop, however slow the diffusion or fast the reaction is. We now bring back the diffusion term, but retain the linearized photon absorption that holds for small values of sc. For first-order
"
# Ai0 l2n 2 2 Un ðXÞ ¼ an Ai Xd ln 0 2 Bi Xd ln Bi ln Here, d ¼ ðDascÞ1=3 and ln ¼ kn =ðDascÞ1=3 are the roots of 0 0 0 0 Ai ðl2 ÞBi ðd l2 Þ Ai ðd l2 ÞBi ðl2 Þ ¼ 0. The constants an are computed from the orthonormality condition hUn ; Un i ¼ 1 and the transform of the initial condition reads Y n ¼ h1; Un i, where the inner product h ; i is defined over x 2 ½0; 1. If sc 1, far enough from the lower asymptote, then the leading term of the summation, for which a0 Y 0 1:41 þ 0:24d2 and l20 d=2, is sufficiently accurate. Using this leading term, the criterion for mass transfer limitations becomes
p Aiðd=2Þ
0 Ai ðd=2Þ 0 Bi ðd=2Þ ¼ 0:95 Biðd=2Þ 0 Bi ðd=2Þ
ð11Þ
which works out to d ¼ 0:84204, or, Dasc ¼ 0:5970. For smaller values of sc, higher reaction rates are possible. This lowers the characteristic reaction time, and for small T the series in Eq. (10) does not converge very fast, such that the faster-decaying higher terms (i.e. with n > 1 and kn np) must be included. A small correction of the leading-term solution,
Daðsc 0:05Þ ¼ 0:5970
ð12Þ
captures the effect of these higher-order terms and has the correct behavior as sc ! 0:05. For scattering media, the expansion of the exponential terms in the rate of photon absorption in Eq. (2) gives a reaction term DaEcA C a ¼ Da ½1 ðscÞ XC a with
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a3 a2 1 x2
ðscÞ ¼ sc ¼ sc a3 þ a2 1x c 1x Da ¼ ½a1 ða2 þ a3 Þc Da ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Da 1 x2
R¼
2 1=2
1=2
1
Da e e dX ¼ C ðTÞ Da expðcsXÞ CðTÞ
ð17Þ
sc
0
which balances the flux rC to the reactive pffiffiffi zone, calculated [10] with an estimated penetration depth of 2 T as
where the expression in terms of x has been obtained using a1 ¼ ½ð1 þ uÞx1 ; a2 ¼ u½1 x þ ð1 x2 Þ
Z
; a3 ¼ 1 þ x þ ð1 x2 Þ
1=2
2 1=2
with u ¼ ½1 þ ð1 x Þ = ½1 þ ð1 x Þ for thin reactors from [7]. Proceeding as for Eq. (8) with these substitutions leads to a similar expression for the concentration difference
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 a a ¼ 3 2 sc ¼ sc 1x a2 þ a3
ð13Þ
ð14Þ
This concludes the analysis of thin reactors. 5. Asymptotic behavior of thick reactors In this section, we develop an approximate solution that demonstrates the important features of reaction and diffusion in stagnant photoreactors, while keeping the mathematics as simple as possible. For thick reactors, light penetrates into only a small fraction of the reactor depth, and the rest of the photoreactor is in the dark. The thickness of the illuminated part is of the order b1 , or in e s1 , where the tilde denotes varidimensionless coordinates, X ables in the illuminated zone. We will proceed by first setting up e ðTÞ, the concentration a balance for this illuminated part to find C in the illuminated part. We ignore the details of the spatial variation of the concentration here and are satisfied with an integral value for the entire illuminated part. We then use this concentration to calculate the concentration in the dark part, collapsing the details of what happens in the small reactive part into a boundary condition for the non-reactive part. Throughout this section, we consider first-order reactions and use Eq. (3) for the rate of photon absorption. The Dahmköhler number then becomes
k½bI0 ð1 qðxÞÞc L2 Da ¼ : D
e 1 CðTÞ pffiffiffiffiffiffiffi pT
ð18Þ
Setting J ¼ R yields the concentration in the reactive zone as
1 Da pffiffiffiffiffiffiffi e CðTÞ ¼ 1þ pT
ð19Þ
sc
As the amount of scattering increases, the relative drop in the rate of photon absorption increases, and mass transfer limitations become important at smaller optical thickness. Note that the lower asymptote is now not at sc ¼ 0:05, but at sc ¼ 0:05ð1 xÞð1 x2 Þ1=2 . The approximate solution for small sc now becomes, upon substitution of Da and ðscÞ into Eq. (12)
! c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1x 1x p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Da sc 0:05 ¼ 0:5970 1x 1 x2
J¼
We can use this expression as a boundary condition for diffusion in a semi-infinite medium. Using a standard self-similarity procedure [10,11], we have
X s1 e e pffiffiffi CðX; TÞ ¼ CðTÞ þ ½1 CðTÞerf 2 T
ð20Þ
which the reader may compare to a numerical solution in Fig. 2. 5.2. Fully developed diffusion throughout the reactor As time progresses, the penetration depth of diffusion advances to the dark end of the reactor, such that the analysis leading to Eq. (19) is not valid beyond that point in time. While an analytical series solution for the finite domain is possible, we prefer here as a simple trial function a parabola that always satisfies the boundary condition of zero flux on the dark end,
PðTÞ 2 e CðX; TÞ ¼ CðTÞ X þ PðTÞX 2
ð21Þ
(a) 1.0
C 0.5
0.0 0.0
Time Increase
0.2
0.4
ð15Þ
0.6
0.8
1.0
0.8
1.0
X
(b) 1.0
and the component balance can be written as 2
@C @ C ¼ Da ecsX C: @T @X 2
ð16Þ
The numerical solution in Fig. 1(d) shows the main characteristics of the evolution of the concentration. At first, the concentration drops in the reactive region but remains unchanged at X ¼ 1, as shown in the figure for a conversion level of 10%. Later, the concentration drops in the entire reactor, until complete conversion.
C 0.5
Time Increase
5.1. Initial development of the diffusion front, T 1 Initially, the concentration CðX; TÞ is unity throughout. At the onset of illumination, the concentration in the reactive part immediately starts to drop such that a gradient in the dark zone develops. The total rate of disappearance of reactant, i.e. reaction rate times volume, is given by
0.0 0.0
0.2
0.4
0.6
X Fig. 2. Numerical (solid) vs. approximate analytical (dashed) solution of the concentration profile for a ¼ c ¼ 1; s ¼ 20 in optically thick photoreactors (a) at high Dahmköhler numbers Da ¼ 1000. (b) At low Dahmköhler number Da ¼ 0:1.
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M. Motegh et al. / Chemical Engineering Journal 247 (2014) 314–319
which has two unknowns, i.e. the concentration in the reactive zone e ðTÞ and the total concentration difference DC ¼ PðTÞ=2. The first C balance that we can use to find these unknowns is that the flux to the reactive zone, dC=dX at X ¼ 0, equals the total rate of disappearance of reactant given by Eq. (17). Upon differentiation of Eq. (21), this yields
dC Da e C ðTÞ ð0; TÞ ¼ PðTÞ ¼ dX sc
ð22Þ
e ðTÞ is obtained from the balance of A differential equation for C the total amount of reaction in the reactive zone with the change in total amount in the dark zone,
d dT
Z 0
1
Da e Da e e CðTÞ þ CðTÞX CðTÞX 2 dX 2sc sc
¼
Da e CðTÞ
sc
which works out to
ðDa þ 3csÞ
e ðTÞ dC e ðTÞ ¼ 3Da C dT
ð23Þ
which is readily integrated with an appropriate initial condition. A smooth transition from Eq. (19) to late diffusion is obtained by switching over at T ¼ ð4pÞ1 , such that the initial condition for Eq. e pÞ1 Þ ¼ ð1 þ Da=2scÞ1 . Integrating and substi(23) is given by Cðð4 tuting in the trial solution Eq. (21) gives as approximate solution for the concentration
CðX; TÞ ¼
1 þ ðDa=2scÞðX 2 2XÞ 3DaT exp 1 þ ðDa=2scÞ Da þ 3sc
ð24Þ
which is sufficiently accurate to calculate the criterion from mass transfer limitations. A small improvement can be made to Eq. (24) by letting the reaction zone have finite length s1 , letting the integration leading up to Eq. (23) have an upper limit ð1 s1 Þ and transforming X ! X s1 . This results in 2
CðX; TÞ ¼
1 þ ðDa=2scÞ½ðX s1 Þ 2ðX s1 Þ 1 þ ðDa=2scÞ 6Das3 T exp 3 ðs 1ÞðDaðs 1Þð2s þ 1Þ þ 6s cÞ
ð25Þ
Fig. 2 shows a comparison of this expression with a numerical solution of Eq. (16) with c ¼ 1; s ¼ 0:1 for large concentration gradients ðDa ¼ 101 Þ and small concentration gradients ðDa ¼ 103 Þ, such that the reader may judge the accuracy of the approximate solution for typical values.
6. Reactors of intermediate optical thickness The previous section have shown asymptotic results for s 1 (optically thin reactors) and for s 1 (optically thick reactors). For kinetic studies, these are the two most relevant cases. Measuring reaction rates in sufficiently thin reactors offers robustness, in the sense that such reactors are not limited by diffusion for any reaction rate. Measuring in thick reactors allows direct calculation of the quantum efficiency from the influx of photons and the reaction rate (provided that one corrects for backscattering), and the simple criterion in Eq. (27) provides a simple check for experimentalists to determine whether the reaction is slow enough – if not, then stirring is needed. In fact, we reiterate here that there are other consequences of unmixed kinetic tests, such as agglomeration of catalyst particles, that make it recommendable to perform such tests always in well-mixed aerated reactors. Perferably, criteria such as Eq. (27) are expressed in terms of measurable quantities – our criterion at first sight seems to have a chicken-and-egg problem: in order to check whether we can measure the kinetics, we need to know the kinetics. However, under conditions where the criterion holds, the value of k observed in an experiment is almost equal to the intrinsic one, and higher intrinsic values of k would have higher observed k values as well. Therefor, Da in Eqs. (14) and (27) can be safely replaced by an ‘‘observed’’ Daobs ¼ kobs ðbI0 Þc L2 =D, which is entirely in directly measurable quantities. For intermediate situations, 1 < sc < 10, there is no approximate solution that allows us to calculate our criterion, and we have to do with numerical solutions. Fig. 3 shows such a calculation for a wide range of optical thickness, with c ¼ 1 and a ¼ 1, i.e. moderate illumination and first-order reaction. All asymptotes are recovered: the maximum Dahmköhler number goes to infinity at sc ¼ 0:05, drops as ðscÞ1 for s 1 and approaches 0:1sc for s > 10. For all of these asymptotes, scattering can included, either using Eq. (3) for thick reactors or by adapting the linearization for thin reactors. We stress here that the intermediate regime is of little importance to kinetic tests. Of course, most commercial-scale reactors will operate at optical thickness close to unity, but such reactors are most either well-mixed, such that the design equations in [7,12] can be used, or they have an intermediate level of mixing that requires full numerical simulations. Finally, we have repeated the numerical calculations of Fig. 3 for 0 < a < 1 to investigate the effect of the reaction order. Again, we have no analytical solutions for these situations. The maximum 100
5.3. Mass transfer limitations in thick reactors Using the approximate solution for developed diffusion (the contribution of the initial development is small enough to ignore for thick reactors) in the criterion for mass transfer limitations, we get
10
1
Da ¼ Da þ 2sc
ð26Þ
from which we conclude that mass transfer is not limiting if Da < 0:1sc, or, in dimensional variables,
k½bI0 ð1 qðxÞÞc < 0:1
Dbc L
ð27Þ
In other words, if a stagnant reactor is doubled in depth, it can only accommodate reactions of half the speed without transport limitations.
0.1 0.01
0.1
1
10
100
Fig. 3. Numerical calculation (full line and markers) of the criterion for mass transfer limitations, plotted as the maximum value of Da vs. sc for a first-order reaction rate ða ¼ 1Þ. The asymptotic values for sc ! 0:05; sc < 1 and sc 1 are included with dotted lines. All results are for exponentially decaying photon absorption (Eq. (1) or (3)).
M. Motegh et al. / Chemical Engineering Journal 247 (2014) 314–319
allowable Dahmköhler number decreased with decreasing a. In fact, a master curve for the entire range could be constructed by replacing Da by Daða þ 1Þ=2, very similar to such corrections in other reaction-diffusion problems. In the range 0 < a < 1, the numerical calculations showed that for non-linear kinetics only this correction is needed to use the criteria as shown in Fig. 3. 7. Conclusions Photon-induced diffusion limitations in photocatalytic reactors have been assessed using an approximate analysis of completely stagnant photoreactors. The key dimensionless parameters were identified to be the Dahmköhler number, optical thickness and to a less extent the scattering albedo. Based on these parameters, a simple closed-form criterion was derived for optically thin and thick reactors, that can help experimentalist to set up their experiments for the determination of kinetics and quantum efficiency. References [1] J. Carberry, Mass diffusion and isothermal catalytic selectivity, Chem. Eng. Sci. 17 (9) (1962) 675–681.
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Chemical Engineering Journal journal homepage: www.elsevier.com/locate/cej
Diffusion limitations in stagnant photocatalytic reactors Mahsa Motegh 1, Jiajun Cen 1, Peter W. Appel, J. Ruud van Ommen, Michiel T. Kreutzer ⇑ Department of Chemical Engineering, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands
h i g h l i g h t s Analyzes of uneven absorption of light in photoreactors vs. counteracting diffusion. Describes the extremes of optically thick reactors and optically thin reactors. Provides a criterion for maximum allowable reaction rate for kinetic experiments.
a r t i c l e
i n f o
Article history: Received 1 November 2013 Received in revised form 13 February 2014 Accepted 19 February 2014 Available online 7 March 2014 Keywords: Photocatalysis Slurry reactors Diffusion limitation Photon absorption rate Mathematical modeling
a b s t r a c t This paper provides a simple criterion to determine when the performance of an unmixed photocatalytic slurry reactor becomes limited by diffusion. We use a 1D description of the reactor and the two-flux intensity model to describe the concentration profile in unmixed photoreactors. We show that the effect of diffusion limitation in optically thick photoreactors is negligible when the Dahmköhler number based on reactor length is smaller than 0:1 sc, where s is the optical thickness and c is the exponent that describes how the reaction rate varies with light intensity. For optically thin reactors, in contrast, we find that the maximum Dahmköhler number scales with the inverse of sc. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction This paper explores the effect of mass-transport limitations on photocatalytic reactors. Invariably, when mass transport is not fast enough to keep up with catalysis, the overall reaction rates changes. Classical examples include the effect of mass transport outside a catalytic particle [1], mass transport inside a catalyst particle [2] and axial dispersion in nearly-plug-flow reactors [3]. For all of these examples, the most useful analysis has been to define a criterion that teaches when change of the rate due to limitations exceeds a given threshold, typically 5%. Such an analysis captures the result of approximate or exact solutions of the governing component balances in a single criterion that can be used by experimentalists. For photocatalytic reactors, such criteria are largely absent. The situation in photocatalysis is more complex: in addition to a nonuniform concentration field c, the optical field variable of relevance, i.e. the rate of photon absorption ea by the particles, is also not constant throughout the reactor. This local volumetric rate of ⇑ Corresponding author. 1
E-mail address: [email protected] (M.T. Kreutzer). Both authors contributed equally.
http://dx.doi.org/10.1016/j.cej.2014.02.075 1385-8947/Ó 2014 Elsevier B.V. All rights reserved.
photon absorption ea appears in the kinetic expression r ¼ f ðc; ea Þ. Gradients in ea lead to gradients in r, which in turn lead to concentration gradients, even if the catalyst is homogeneously dispersed and particle-level gradients can be ignored. Of course, vigorous stirring can eliminate such concentration gradients, but we find many examples [e.g. 4,5] of unstirred catalytic performance tests that lack forced convection (e.g., stirring) or reported natural convection (e.g., due to heating by incoming light or electron/hole recombination). Herein, we analyze the simplest case of mass transport limitations in such stagnant photoreactors. To keep the problem tractable, we consider a reactor volume between parallel transparent plates, with light of intensity I0 entering perpendicularly onto one side, and we use a two-flux model for the radiation transfer, as discussed below. This renders the problem one-dimensional. After a brief formulation of the problem, we define our five-percent criterion for the onset of mass transport limitations. We then explore limiting cases of optically very thin and optically very thick reactors. This gives analytical expressions for the asymptotes of the criterion, which have the most relevance in catalyst performance tests, and we use numerical calculations for the intermediate regime that is more important in design of efficient photoreactors. We mention here that there are other aspects of
M. Motegh et al. / Chemical Engineering Journal 247 (2014) 314–319
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unmixed catalyst testing, such as aggregation and sedimentation of catalyst particles, concurrent possible diffusion limitation of oxygen from the air, and the likely natural convection from heating on the illuminated side, that are beyond the scope of this paper.
pffiffiffiffiffi r / ea when electron–hole pairs are generated much faster than they can migrate to the surface. Here, we shall use a power-law a expression r ¼ kc eca , where the kinetic constant k can be any value and the kinetic constants 0 < a 6 1 and 0:5 < c < 1, typically.
2. Problem formulation
2.3. Governing component balance
2.1. Local rate of photon absorption
Without convection, the transient component balance for concentration cðx; tÞ only has a diffusion term and a reaction term:
We first describe the local rate of photon absorption inside the slurry. The simplest analysis just uses the Lambert–Beer equation that ignores all scattering, resulting in an exponential decay of the light intensity I inside the reactor, IðxÞ ¼ I0 expðbxÞ. The local rate of photon absorption is then given by
@c @2c c ¼ D 2 kea ca ; @t @x
ea ¼ dI=dx ¼ bI0 expðbxÞ:
ð1Þ
where b is the extinction coefficient and x is the spatial coordinate that varies from x ¼ 0 where the light enters to x ¼ L on the other end. A useful parameter is the optical thickness s ¼ bL. A reactor is thick (opaque) if s 1, indicating that very little light penetrates through the reactor. Conversely, a reactor is optically thin if s 1. Photocatalytic particles not only absorb light but also scatter it. Scattering occurs in all directions, and the full calculation of the radiation transfer equation is difficult. The two-flux model [6] simplifies the complex analysis of radiation transfer by assuming that light only travels in one direction. Light can be absorbed in a differential slice of thickness dx, or it can be scattered back, or it can pass through without changing path. The last option occurs because of a forward-scattering event or because the photon simply does not interact with any particle. Of course, backscattered light can be scattered again. For unidirectional light that enters the slurry at x ¼ 0, this results in
bx
ea ¼ bI0 a1 a2 e þ a3 e
bx
ð2Þ
where and a1 ; a2 ; a3 are dimensionless coefficients. The term a2 expðbxÞ in Eq. (2) complicates the analysis, but it turns out that for thick reactors, a2 is vanishingly small, such that we can write a pseudo-Lambert–Beer expression that consists only of an exponential decay term. Motegh et al. [7] showed that for optically thick photoreactors,
ea ¼ bI0 ½1 qðxÞebx
ð3Þ 1 2 1=2
where qðxÞ ¼ x½1 þ ð1 x Þ is the fraction of lost photons via backscattering out of the reactor. It depends on the scattering albedo x ¼ r=b, where r is a scattering coefficient. Limiting values of x are x ¼ 0 when particles do not scatter but only absorb and x ¼ 1 when particles only scatter but do not absorb light. 2.2. Kinetics To use the local rate of photon absorption in a reactor model, we need a kinetic expression that relates the absorption of photons by particles to chemical conversion at the particle surface. Many such expressions have been proposed. Based on microkinetics, one can derive expressions for the reaction rate per unit slurry volume r 1=2 of the form r ¼ k1 f ðcÞ½1 þ ð1 þ k2 ea Þ , where k1 and k2 are kinetic constants and f ðcÞ 2 ½0; 1 describes the surface saturation as a function of concentration c [8]. Most relevant photocatalytic processes are pollutant removals at low concentrations, for which surface saturation does not occur, such that the reaction is observed to be first order. At higher concentration, the observed reaction order can be smaller. The term in the square root describes the effect of electron–hole recombination. When ea is small, electron– hole pairs migrate faster to the surface than they are generated, such that recombination is minimal and r / ea . Conversely,
ð4Þ
with initial condition cðx; 0Þ ¼ c0 and no-flux boundary conditions @c=@xð0; tÞ ¼ @c=@xðL; tÞ ¼ 0. Note that r and ea are defined per unit volume of the slurry mixture, such that the catalyst concentration is ‘‘hidden’’ in k. We are interested in the concentration profile in the entire reactor, so we scale length as X ¼ x=L. The concentration falls from the initial value c0 to zero, so we have for dimensionless concentration C ¼ c=c0 . The characteristic time iis either the diffusion h 1 a1 time L2 =D or the reaction time kc0 ðbI0 Þc . The ratio of these characteristic times is the Dahmköhler number a1
Da ¼
kc0 ðbI0 Þc L2 : D
ð5Þ
The significance of Da is shown in Fig. 1: small values, in Fig. 1(a) and (b), indicate that gradients are small, whereas for Da 1 in Fig. 1(c) and (d), gradients in concentration are significant. We scale 1 time with the diffusion time, i.e. T ¼ ðL2 =DÞ t, and obtain
@C @ 2 C ¼ DaEca C a : @T @X 2
ð6Þ
where Ea ¼ ea =bI0 is the dimensionless rate of photon absorption. The scaled initial condition and boundary conditions are CðX; 0Þ ¼ 1; @C=@Xð0; TÞ ¼ @C=@Xð1; TÞ ¼ 0. 3. Criterion for mass transfer limitations A good instantaneous measure for diffusion limitations is the difference in concentration between the illuminated end and the dark end of the reactor. We define a time-averaged version over the entire conversion as
R1
¼
0
½Cð1; TÞ Cð0; TÞdT R1 Cð1; TÞdT 0
ð7Þ
and define as a criterion for the absence of mass transport limitations < 0:05. As discussed above, we limit ourselves to 0 < a 6 1. As is generally the case, for first-order reactions the concentration asymptotically approaches zero, whereas for fractionalorder reactions ð0 < a < 1Þ the concentration becomes zero in finite time. From numerical solutions of Eq. (6) where non-negativity of concentration was ensured, we found for first-order reactions that we could truncate the integration at T 10, with negligible change in upon continued integration. For fractional-order reactions, reaches a finite value when the concentration becomes zero everywhere, which happens in finite time. In other words, never diverges to infinitely for the range of a that we are interested in and gives a good time-averaged measure for the extent of diffusion limitations. The criterion ¼ 0:05 will depend on the Dahmköhler number Da, the optical thickness s, the scattering albedo x and the reaction orders a and c. The problem is well tractable for the first-order reactions without scattering, i.e. the case a ¼ 1; x ¼ 0, which we will describe in detail. This will reveal the important features of the boundary, which we write as ðDa; scÞ ¼ 0:05, because it will turn out sc always appear together as a group. Subsequently, we
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Increase in
X=τ -1
(a) 1.0 C
0.5
0.1
0.1
0.5
0.5
rxn region
rxn region 0.75
0.0
0.75
Ea
dark region
(d)
(c) 1.0
0.1
0.1
0.5
Ea
C
Increase in Da
Ea
-1 (b) X=τ
0.5
0.5
0.75
rxn region
0.75
rxn region
0.0 0
1
Ea dark region
0
1
X
X
Fig. 1. (a–d) The dimensionless concentration profile as a function of depth X when the average conversion is set to 10%, 50%, and 75% (solid lines) and the profile of the rate of photon absorption (dashed lines). (a and b) Low Dahmköhler number ðDa ¼ 0:1Þ for (a) optically moderately thin ðs ¼ 1Þ and (b) optically thick ðs ¼ 10Þ case. (c and d) High Dahmköhler number ðDa ¼ 1000Þ for (c) a optically moderately thin ðs ¼ 1Þ and (d) optically thick ðs ¼ 10Þ case.
shall see that using Eq. (3) permits the use of the zero-scattering result for thick reactors with scattering, such that only for sc 1 we need to use numerical calculations to describe situations where x > 0, i.e., situations with scattering in thin reactors. Finally, we will discuss numerical solutions for the situation a < 1.
CðX; TÞ ¼
4. Asymptotic behavior for thin reactors
ð8Þ
which is readily solved by separation of variables, resulting in CðX; TÞ ¼ exp½Dað1 scXÞT for first-order reactions and CðX; TÞ ¼ ½1 DaTð1 scXÞð1 aÞ1=ð1aÞ for fractional-order reactions. Substituting either of these into Eq. (7) and integrating until (for a < 1) the concentration becomes zero reveals that, for any reaction order 0 < a 1, irrespective of the value of the Dahmköhler number,
¼ sc:
1 X Y n exp k2n þ Dað1 scÞ T Un ðXÞ;
ð10Þ
n¼0
In thin reactors, such as shown in Fig. 1(a) and (c), we have nonzero photon absorption throughout the reactor. The worst-case scenario, as far as diffusion limitations go, is that diffusion is not able at all to smooth out the gradients caused by non-homogeneous reaction. This occurs at high Da, such that the ð@ 2 C=@X 2 Þterm is negligible with respect to the reaction term. Thus, a highDa situation can be approximately analyzed by dropping the diffusion term, such that the concentration is simply given by @C=@T ¼ DaEca C a . For non-scattering photon-absorption (Eq. (1)) in thin reactors, such that sX 1, this simplifies to
@C=@T ¼ Dað1 scXÞC a ;
reactions, a solution of the resulting balance can be expressed in terms of Airy functions. Using the finite Fourier transform method [9, chap. 5], a solution, in the form of an infinite sum using orthonormal basis functions Un ðXÞ, is readily found to be
ð9Þ
Thus, for thin reactors, the asymptotic behavior of the criterion for mass transfer limitations is that as sc goes to 0.05, Da may go to infinity. For sc < 0:05, the difference in the rate of photon absorption is so small that significant gradients in concentration across the reactor can never develop, however slow the diffusion or fast the reaction is. We now bring back the diffusion term, but retain the linearized photon absorption that holds for small values of sc. For first-order
"
# Ai0 l2n 2 2 Un ðXÞ ¼ an Ai Xd ln 0 2 Bi Xd ln Bi ln Here, d ¼ ðDascÞ1=3 and ln ¼ kn =ðDascÞ1=3 are the roots of 0 0 0 0 Ai ðl2 ÞBi ðd l2 Þ Ai ðd l2 ÞBi ðl2 Þ ¼ 0. The constants an are computed from the orthonormality condition hUn ; Un i ¼ 1 and the transform of the initial condition reads Y n ¼ h1; Un i, where the inner product h ; i is defined over x 2 ½0; 1. If sc 1, far enough from the lower asymptote, then the leading term of the summation, for which a0 Y 0 1:41 þ 0:24d2 and l20 d=2, is sufficiently accurate. Using this leading term, the criterion for mass transfer limitations becomes
p Aiðd=2Þ
0 Ai ðd=2Þ 0 Bi ðd=2Þ ¼ 0:95 Biðd=2Þ 0 Bi ðd=2Þ
ð11Þ
which works out to d ¼ 0:84204, or, Dasc ¼ 0:5970. For smaller values of sc, higher reaction rates are possible. This lowers the characteristic reaction time, and for small T the series in Eq. (10) does not converge very fast, such that the faster-decaying higher terms (i.e. with n > 1 and kn np) must be included. A small correction of the leading-term solution,
Daðsc 0:05Þ ¼ 0:5970
ð12Þ
captures the effect of these higher-order terms and has the correct behavior as sc ! 0:05. For scattering media, the expansion of the exponential terms in the rate of photon absorption in Eq. (2) gives a reaction term DaEcA C a ¼ Da ½1 ðscÞ XC a with
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a3 a2 1 x2
ðscÞ ¼ sc ¼ sc a3 þ a2 1x c 1x Da ¼ ½a1 ða2 þ a3 Þc Da ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Da 1 x2
R¼
2 1=2
1=2
1
Da e e dX ¼ C ðTÞ Da expðcsXÞ CðTÞ
ð17Þ
sc
0
which balances the flux rC to the reactive pffiffiffi zone, calculated [10] with an estimated penetration depth of 2 T as
where the expression in terms of x has been obtained using a1 ¼ ½ð1 þ uÞx1 ; a2 ¼ u½1 x þ ð1 x2 Þ
Z
; a3 ¼ 1 þ x þ ð1 x2 Þ
1=2
2 1=2
with u ¼ ½1 þ ð1 x Þ = ½1 þ ð1 x Þ for thin reactors from [7]. Proceeding as for Eq. (8) with these substitutions leads to a similar expression for the concentration difference
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2 a a ¼ 3 2 sc ¼ sc 1x a2 þ a3
ð13Þ
ð14Þ
This concludes the analysis of thin reactors. 5. Asymptotic behavior of thick reactors In this section, we develop an approximate solution that demonstrates the important features of reaction and diffusion in stagnant photoreactors, while keeping the mathematics as simple as possible. For thick reactors, light penetrates into only a small fraction of the reactor depth, and the rest of the photoreactor is in the dark. The thickness of the illuminated part is of the order b1 , or in e s1 , where the tilde denotes varidimensionless coordinates, X ables in the illuminated zone. We will proceed by first setting up e ðTÞ, the concentration a balance for this illuminated part to find C in the illuminated part. We ignore the details of the spatial variation of the concentration here and are satisfied with an integral value for the entire illuminated part. We then use this concentration to calculate the concentration in the dark part, collapsing the details of what happens in the small reactive part into a boundary condition for the non-reactive part. Throughout this section, we consider first-order reactions and use Eq. (3) for the rate of photon absorption. The Dahmköhler number then becomes
k½bI0 ð1 qðxÞÞc L2 Da ¼ : D
e 1 CðTÞ pffiffiffiffiffiffiffi pT
ð18Þ
Setting J ¼ R yields the concentration in the reactive zone as
1 Da pffiffiffiffiffiffiffi e CðTÞ ¼ 1þ pT
ð19Þ
sc
As the amount of scattering increases, the relative drop in the rate of photon absorption increases, and mass transfer limitations become important at smaller optical thickness. Note that the lower asymptote is now not at sc ¼ 0:05, but at sc ¼ 0:05ð1 xÞð1 x2 Þ1=2 . The approximate solution for small sc now becomes, upon substitution of Da and ðscÞ into Eq. (12)
! c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1x 1x p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Da sc 0:05 ¼ 0:5970 1x 1 x2
J¼
We can use this expression as a boundary condition for diffusion in a semi-infinite medium. Using a standard self-similarity procedure [10,11], we have
X s1 e e pffiffiffi CðX; TÞ ¼ CðTÞ þ ½1 CðTÞerf 2 T
ð20Þ
which the reader may compare to a numerical solution in Fig. 2. 5.2. Fully developed diffusion throughout the reactor As time progresses, the penetration depth of diffusion advances to the dark end of the reactor, such that the analysis leading to Eq. (19) is not valid beyond that point in time. While an analytical series solution for the finite domain is possible, we prefer here as a simple trial function a parabola that always satisfies the boundary condition of zero flux on the dark end,
PðTÞ 2 e CðX; TÞ ¼ CðTÞ X þ PðTÞX 2
ð21Þ
(a) 1.0
C 0.5
0.0 0.0
Time Increase
0.2
0.4
ð15Þ
0.6
0.8
1.0
0.8
1.0
X
(b) 1.0
and the component balance can be written as 2
@C @ C ¼ Da ecsX C: @T @X 2
ð16Þ
The numerical solution in Fig. 1(d) shows the main characteristics of the evolution of the concentration. At first, the concentration drops in the reactive region but remains unchanged at X ¼ 1, as shown in the figure for a conversion level of 10%. Later, the concentration drops in the entire reactor, until complete conversion.
C 0.5
Time Increase
5.1. Initial development of the diffusion front, T 1 Initially, the concentration CðX; TÞ is unity throughout. At the onset of illumination, the concentration in the reactive part immediately starts to drop such that a gradient in the dark zone develops. The total rate of disappearance of reactant, i.e. reaction rate times volume, is given by
0.0 0.0
0.2
0.4
0.6
X Fig. 2. Numerical (solid) vs. approximate analytical (dashed) solution of the concentration profile for a ¼ c ¼ 1; s ¼ 20 in optically thick photoreactors (a) at high Dahmköhler numbers Da ¼ 1000. (b) At low Dahmköhler number Da ¼ 0:1.
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which has two unknowns, i.e. the concentration in the reactive zone e ðTÞ and the total concentration difference DC ¼ PðTÞ=2. The first C balance that we can use to find these unknowns is that the flux to the reactive zone, dC=dX at X ¼ 0, equals the total rate of disappearance of reactant given by Eq. (17). Upon differentiation of Eq. (21), this yields
dC Da e C ðTÞ ð0; TÞ ¼ PðTÞ ¼ dX sc
ð22Þ
e ðTÞ is obtained from the balance of A differential equation for C the total amount of reaction in the reactive zone with the change in total amount in the dark zone,
d dT
Z 0
1
Da e Da e e CðTÞ þ CðTÞX CðTÞX 2 dX 2sc sc
¼
Da e CðTÞ
sc
which works out to
ðDa þ 3csÞ
e ðTÞ dC e ðTÞ ¼ 3Da C dT
ð23Þ
which is readily integrated with an appropriate initial condition. A smooth transition from Eq. (19) to late diffusion is obtained by switching over at T ¼ ð4pÞ1 , such that the initial condition for Eq. e pÞ1 Þ ¼ ð1 þ Da=2scÞ1 . Integrating and substi(23) is given by Cðð4 tuting in the trial solution Eq. (21) gives as approximate solution for the concentration
CðX; TÞ ¼
1 þ ðDa=2scÞðX 2 2XÞ 3DaT exp 1 þ ðDa=2scÞ Da þ 3sc
ð24Þ
which is sufficiently accurate to calculate the criterion from mass transfer limitations. A small improvement can be made to Eq. (24) by letting the reaction zone have finite length s1 , letting the integration leading up to Eq. (23) have an upper limit ð1 s1 Þ and transforming X ! X s1 . This results in 2
CðX; TÞ ¼
1 þ ðDa=2scÞ½ðX s1 Þ 2ðX s1 Þ 1 þ ðDa=2scÞ 6Das3 T exp 3 ðs 1ÞðDaðs 1Þð2s þ 1Þ þ 6s cÞ
ð25Þ
Fig. 2 shows a comparison of this expression with a numerical solution of Eq. (16) with c ¼ 1; s ¼ 0:1 for large concentration gradients ðDa ¼ 101 Þ and small concentration gradients ðDa ¼ 103 Þ, such that the reader may judge the accuracy of the approximate solution for typical values.
6. Reactors of intermediate optical thickness The previous section have shown asymptotic results for s 1 (optically thin reactors) and for s 1 (optically thick reactors). For kinetic studies, these are the two most relevant cases. Measuring reaction rates in sufficiently thin reactors offers robustness, in the sense that such reactors are not limited by diffusion for any reaction rate. Measuring in thick reactors allows direct calculation of the quantum efficiency from the influx of photons and the reaction rate (provided that one corrects for backscattering), and the simple criterion in Eq. (27) provides a simple check for experimentalists to determine whether the reaction is slow enough – if not, then stirring is needed. In fact, we reiterate here that there are other consequences of unmixed kinetic tests, such as agglomeration of catalyst particles, that make it recommendable to perform such tests always in well-mixed aerated reactors. Perferably, criteria such as Eq. (27) are expressed in terms of measurable quantities – our criterion at first sight seems to have a chicken-and-egg problem: in order to check whether we can measure the kinetics, we need to know the kinetics. However, under conditions where the criterion holds, the value of k observed in an experiment is almost equal to the intrinsic one, and higher intrinsic values of k would have higher observed k values as well. Therefor, Da in Eqs. (14) and (27) can be safely replaced by an ‘‘observed’’ Daobs ¼ kobs ðbI0 Þc L2 =D, which is entirely in directly measurable quantities. For intermediate situations, 1 < sc < 10, there is no approximate solution that allows us to calculate our criterion, and we have to do with numerical solutions. Fig. 3 shows such a calculation for a wide range of optical thickness, with c ¼ 1 and a ¼ 1, i.e. moderate illumination and first-order reaction. All asymptotes are recovered: the maximum Dahmköhler number goes to infinity at sc ¼ 0:05, drops as ðscÞ1 for s 1 and approaches 0:1sc for s > 10. For all of these asymptotes, scattering can included, either using Eq. (3) for thick reactors or by adapting the linearization for thin reactors. We stress here that the intermediate regime is of little importance to kinetic tests. Of course, most commercial-scale reactors will operate at optical thickness close to unity, but such reactors are most either well-mixed, such that the design equations in [7,12] can be used, or they have an intermediate level of mixing that requires full numerical simulations. Finally, we have repeated the numerical calculations of Fig. 3 for 0 < a < 1 to investigate the effect of the reaction order. Again, we have no analytical solutions for these situations. The maximum 100
5.3. Mass transfer limitations in thick reactors Using the approximate solution for developed diffusion (the contribution of the initial development is small enough to ignore for thick reactors) in the criterion for mass transfer limitations, we get
10
1
Da ¼ Da þ 2sc
ð26Þ
from which we conclude that mass transfer is not limiting if Da < 0:1sc, or, in dimensional variables,
k½bI0 ð1 qðxÞÞc < 0:1
Dbc L
ð27Þ
In other words, if a stagnant reactor is doubled in depth, it can only accommodate reactions of half the speed without transport limitations.
0.1 0.01
0.1
1
10
100
Fig. 3. Numerical calculation (full line and markers) of the criterion for mass transfer limitations, plotted as the maximum value of Da vs. sc for a first-order reaction rate ða ¼ 1Þ. The asymptotic values for sc ! 0:05; sc < 1 and sc 1 are included with dotted lines. All results are for exponentially decaying photon absorption (Eq. (1) or (3)).
M. Motegh et al. / Chemical Engineering Journal 247 (2014) 314–319
allowable Dahmköhler number decreased with decreasing a. In fact, a master curve for the entire range could be constructed by replacing Da by Daða þ 1Þ=2, very similar to such corrections in other reaction-diffusion problems. In the range 0 < a < 1, the numerical calculations showed that for non-linear kinetics only this correction is needed to use the criteria as shown in Fig. 3. 7. Conclusions Photon-induced diffusion limitations in photocatalytic reactors have been assessed using an approximate analysis of completely stagnant photoreactors. The key dimensionless parameters were identified to be the Dahmköhler number, optical thickness and to a less extent the scattering albedo. Based on these parameters, a simple closed-form criterion was derived for optically thin and thick reactors, that can help experimentalist to set up their experiments for the determination of kinetics and quantum efficiency. References [1] J. Carberry, Mass diffusion and isothermal catalytic selectivity, Chem. Eng. Sci. 17 (9) (1962) 675–681.
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