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Ind. Eng. Chem. Res. 1996, 35, 712-720
Effectiveness Factors for Photocatalytic Reactions Occurring in Planar Membranes Morgan E. Edwards,†,‡,§ Carlos M. Villa,§ Charles G. Hill, Jr.,*,§ and Thomas W. Chapman§ Ink Jet Media Operation, Hewlett-Packard Company, 16399 West Bernardo Drive, San Diego, California 92127-1899, and Department of Chemical Engineering, University of Wisconsin, 1415 Engineering Drive, Madison, Wisconsin 53706
Effectiveness factors for porous catalysts have been determined for photocatalytic reactions characterized by integer order and by generalized Langmuir-Hinshelwood/Hougen-Watson rate expressions. The geometry of interest is a planar porous membrane that is illuminated from either the front or the rear by a planar light source. Effectiveness factors for these systems depend not only on Thiele moduli defined herein but also on the product of the effective membrane thickness and the optical absorption coefficient of the porous membrane. Introduction In both the mathematical treatment of kinetic data obtained for heterogeneous catalytic reactions and the design of reactors for carrying out such reactions on an industrial scale, an important consideration is the interaction between mass transport processes and the reactions that occur on the surface of the catalyst. Analyses of such interactions and their implications with respect to potential mass transfer limitations on reaction rates constitute an important body of chemical engineering literature. The problem of intraparticle mass transport in conventional heterogeneous catalysts has been treated by a number of researchers, beginning with Damko¨hler (1937) in Germany, Thiele (1939) in the United States, and Zeldovitch (1939) in the USSR. In the intervening half-century, innumerable articles have extended this work to encompass a wide variety of catalyst geometries, reaction rate expressions, nonisothermal systems, and competitive (parallel) and consecutive (series) reactions, as well as the effects of deactivation of the catalyst. Comprehensive treatises by Aris (1975) and Jackson (1977) provide excellent summaries of research in this area, while more concise treatments are available in the texts by Smith (1981), Hill (1977), Carberry (1976), and Fogler (1992). The solutions to problems of this type are usually expressed in terms of catalyst effectiveness factors, i.e., the ratio of the rate actually observed to that which would be observed in the absence of concentration and temperature gradients within the catalyst. The purpose of this paper is to present the results of an effectiveness factor analysis of reactions taking place within photocatalytic membranes, thereby extending the previous analyses which have been limited to reactions which do not require the presence of photons of an appropriate wavelength. This study was stimulated by ongoing research concerning the use of photocatalytic oxidation processes for the remediation of both aqueous and gaseous streams containing hazardous organic compounds, especially chlorinated species (see, for example, Sabate et al. (1991, 1992b)). * Author to whom correspondence should be addressed. E-mail address:
[email protected]. † Hewlett-Packard Co. ‡ E-mail address:
[email protected]. § University of Wisconsin.
0888-5885/96/2635-0712$12.00/0
Problem Description. Consider a porous thin film of a photocatalyst (e.g., TiO2) supported on an impermeable, nonreflecting support (see Figure 1). The characteristics of the catalyst layer are assumed to be similar to those of the thin titania films used in previous investigations in our laboratory (Aguado and Anderson, 1993; Sabate et al., 1991, 1992a,b; Zeltner et al., 1993). Because the films normally employed for photocatalysis are thin compared to their principal radii of curvature, it is appropriate to treat them as flat plates. The surface of the photocatalyst is illuminated from one side or the other by a source of visible or ultraviolet radiation that provides uniform intensity to the entire exposed face of the catalyst. The pores are regarded as having a straight cylindrical geometry with axes perpendicular to the plane of the film. No significant temperature gradients exist within the catalyst. In addition, we assume that the reaction mixtures of interest are sufficiently dilute that they may be treated as if there is negligible convective transport of reactants resulting from diffusional transport, i.e., as if the transport can be regarded as mathematically equivalent to equimolar counterdiffusion of reactant and product species within the membrane structure. Under steady-state conditions, the one-dimensional continuity equation is
D
d2CS 2
dx
)
()
2 r rj
(1)
where CS is the concentration of species S in the fluid phase, x is the length coordinate parallel to the direction of the incident radiation (i.e., measured perpendicular to the film), rj is the average pore radius, r is the reaction rate per unit surface area of the photocatalyst, and D is the diffusion coefficient of species S. When the catalyst slab is illuminated from the solution side (see Figure 1), eq 1 is subject to the following boundary conditions:
CS ) CS0
at x ) 0
(2)
dCS )0 dx
at x ) L
(3)
where L is the average length of a pore. When the catalyst slab is illuminated from the support side, the following boundary conditions apply: © 1996 American Chemical Society
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 713
illuminated from the support side are
C)1
at X ) 1
(10)
dC )0 dX
at X ) 0
(11)
For this configuration, r0 corresponds to the reaction rate at X ) 1. We have considered both simple integer-order rate expressions of the form
r ) kφCnS
(12)
(where n is the order of the reaction with respect to the concentration of the reactant) and more complex rate expressions (see below). The rate expressions considered here are more complicated than those treated previously because optical absorption by the catalyst film causes the photon flux (φ) to vary with position according to the relation
φ ) φ0e-µx
Figure 1. Schematic diagram of a photocatalytic system which is illuminated from the solution side of the catalyst film.
CS ) CS0
at x ) L
(4)
dCS )0 dx
at x ) 0
(5)
where x ) 0 now corresponds to the photocatalyst/ support interface and x ) L corresponds to the fluid/ photocatalyst interface. It is customary to rewrite eqs 1-5 in dimensionless form. In our particular case, it is convenient to employ the following definition of the general Thiele modulus (Λ):
2L2r0 Λ ≡ DCS0rj 2
(6)
where r0 is the reaction rate per unit area evaluated at the surface of the catalyst exposed to the fluid medium, regardless of whether the incident radiation strikes the solution side or the support side of the photocatalyst film. Thus, for the system of interest, eq 1 can be written as
d2C ) Λ2r* dX2
(7)
where C ) CS/CS0, X ) x/L, and r* ) r/r0. For the case where the photocatalyst is illuminated from the solution side, the corresponding dimensionless forms of the boundary conditions are
C)1
at X ) 0
(8)
dC )0 dX
at X ) 1
(9)
For this geometry, r0 corresponds to the reaction rate at X ) 0. The corresponding dimensionless forms of the boundary conditions for the case where the photocatalyst is
(13)
where φ0 is the uniform flux of photons at the surface of the photocatalytic film exposed to the radiation, x is the penetration depth of photons into the catalyst coating, and µ is the absorption coefficient per unit thickness of the film. The value of the absorption coefficient depends on both the chemical nature of the solid and the morphology (especially the porosity) of the membranes, as well as the frequency of the incident radiation. Several researchers have reported that photocatalytic oxidation reactions obey Langmuir-Hinshelwood/Hougen-Watson (LHHW) rate expressions (Al-Ekabi and Serpone, 1988; Al-Ekabi et al., 1989; Okamoto et al., 1985; Sabate et al., 1991; Turchi and Ollis, 1990). However, questions persist as to the mechanisms involved. Turchi and Ollis (1990) have indicated that several plausible mechanisms yield similar rate expressions. For present purposes, we have elected to employ a semiempirical rate expression for the oxidation of an organic substrate by oxygen. This expression is similar in mathematical form to a LHHW rate expression for a single site mechanism, viz.,
r ) kφKSCS/(1 + KO2CO2 + KSCS)
(14)
where r is the reaction rate expressed per unit surface area, CS and CO2 are the concentrations of reactant and oxygen, respectively, k is the reaction rate constant, and KO2 and KS are the adsorption equilibrium constants for oxygen and the reactant species, respectively. Numerical Solutions of the Differential Transport Equation. After the rate expressions given by eqs 12 and 14 are substituted into eq 7, the resulting differential equations can be solved numerically using orthogonal collocation. This approach has been employed extensively in modeling a variety of chemical engineering systems; e.g., see Villadsen and Stewart (1967) and Wang and Stewart (1983). In this approach the desired concentration profile inside the catalyst is approximated in terms of a truncated series of orthogonal polynomials. In the present study, the family of shifted Legendre polynomials was used to represent the concentration values at selected points (known as collocation points) within the membrane. Once the concentrations at the selected collocation points have been
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determined for a particular rate expression, the effectiveness factor of the photocatalyst is readily calculated by quadrature of the local reaction rate. In general, the effectiveness factor for a catalyst is defined in the following manner (Hill, 1977):
For a system which is illuminated from the solution side and obeys a zero-order rate expression, eq 7 becomes
actual rate for the entire catalyst rate evaluated at external surface conditions (15)
Integration of eq 19 subject to eqs 8 and 9 gives the concentration profile as
η≡
For purposes of the present paper, the external surface is taken as that exposed to the fluid containing the reactants. Thus, for the geometry of interest
∫0 2πrjr dx ∫0 r dx L
η≡
L
2πrjLr0
)
Lr0
(16)
When the photocatalyst is illuminated from the support side, this definition can lead to effectiveness factors in excess of unity. However, for illumination from the side exposed to the fluid medium, the values of the effectiveness factor will be less than or equal to unity. In the solution for the effectiveness factor as a function of the generalized Thiele modulus, it is appropriate to introduce another dimensionless parameter, the optical film thickness (∆), defined as the product of the membrane thickness and the absorptivity per unit length
∆ ≡ µL
C)1-
(18)
Values of the effectiveness factor were determined for values of ∆ ranging from zero (no attenuation of the incident radiation) to 10 (where the intensity of the radiation emerging from the membrane is less than 0.005% of the intensity of the radiation incident on the membrane). Results of the Effectiveness Factor Calculations. The numerical solutions to both the differential equation of interest (eq 7) and the associated effectiveness factor calculation yield functional relationships between this factor, the Thiele modulus, and the optical density of the membrane. The Thiele modulus itself depends explicitly on the variables given in eq 6 and implicitly (via r0) on the effective flux of radiation and (for rate expressions that are other than first-order) on the solute concentration at the exterior surface of the membrane.
Λ2 [1 - e-∆X - ∆Xe-∆] ∆2
This section presents the results obtained for four rate expressions: zero-, first-, and second-order kinetics, and a single-site Langmuir-Hinshelwood/Hougen-Watson rate expression. Zero-Order Kinetics. For zero-order kinetics, it is possible to obtain a closed-form solution to eq 7 which describes the dimensionless concentration profile of the organic substrate in the pore. By contrast, the other three rate expressions will require the use of collocation. To indicate the general procedure involved in determining the effectiveness factor from a concentration profile, we shall begin by considering the zero-order situation.
(20)
Λ2 1 e 2 -∆ ∆ [1 - e - ∆e-∆]
(21)
This inequality serves to define a critical condition separating the regime in which the concentration of substrate is finite throughout the pore length from that in which this concentration goes to zero at some critical (dimensionless) distance from the pore mouth, namely, Xc. For a fixed value of ∆, a critical value of the Thiele modulus, Λc, can be determined from inequality 21 as
Λc2 )
∆2 [1 - e-∆ - ∆e-∆]
(22)
For values of Λ e Λc, the concentration of reactant is finite throughout the pore length, and eq 16 for the effectiveness factor becomes
η)
∫01r dX ∫01kφ0e-∆X dX r0
kφ0
)
)
∫01e-∆X dX
(23)
where we have employed the definition of the dimensionless coordinate for position, X ) x/L. Integration gives
η)
1 - e-∆ ∆
(24)
For values of Λ in excess of the critical value, the concentration of substrate becomes zero at some critical distance Xc from the pore mouth. For this situation, the boundary condition given by eq 9 must be replaced by the following condition
C)0 Effectiveness Factors for Cases Where the Catalyst Is Illuminated from the Solution Side (Figure 1)
(19)
Since the concentration of no species can take on a negative value, eq 20 is subject to the constraint that C g 0. Further physical constraints require that 0 e C e 1. Hence, eq 20 is applicable only when
(17)
It is now possible to rewrite eq 13 in terms of the optical film thickness.
φ ) φ0e-∆X
d2C ) Λ2e-∆X dX2
at X ) Xc
(25)
For this case, the concentration profile for 0 e X e Xc is given by
C)1-
[
X Λ2 -∆X X + e - 1 + (1 - e-∆Xc) Xc ∆2 Xc
]
(26)
while the expression for the effectiveness factor becomes
∫0X kφ0e-∆X dX (1 - e-∆X ) η) ) ∆ ∫01kφ0 dX c
c
(27)
where the critical distance Xc can be determined from the requirement that the dimensionless concentration
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 715
Figure 2. Plots of effectiveness factor versus Thiele modulus for a reaction which obeys a zero-order rate expression. The catalyst is illuminated from the solution side.
Figure 3. Plots of effectiveness factor versus Thiele modulus for a reaction which obeys a first-order rate expression. The catalyst is illuminated from the solution side.
gradient must also be zero at X ) Xc. Thus at Xc
∆2 ) [1 - e-∆Xc - ∆Xce-∆Xc] Λ2
(28)
Numerical solutions to eq 28 have been obtained for various combinations of ∆ and Λ. These results were then substituted into eq 27 to generate the effectiveness factor plots shown in Figure 2. For values of the Thiele modulus less than the critical value, Λc, the effectiveness factor is constant at a value determined by ∆. At larger values of Λ, the effectiveness factor decreases with increasing Thiele modulus because the substrate concentration goes to zero at increasingly shorter distances from the mouth of the pore. First- and Second-Order Kinetics. If the reaction of interest obeys first-order kinetics or second-order kinetics, then eq 7 becomes
d2C/dX2 ) Λ2Ce-∆X
first order
(29)
second order
(30)
or
d2C/dX2 ) Λ2C2e-∆X
These equations may be solved using orthogonal collocation to determine the concentration profiles. Equation 16 may then be used to determine the effectiveness factor via numerical integration of the appropriate integral. Corresponding plots of the effectiveness factor versus Thiele modulus for several values of ∆ are shown in Figures 3 and 4. LHHW Rate Expressions. We have also calculated the effectiveness factors for photocatalytic systems which obey rate expressions of the LHHW form given
Figure 4. Plots of effectiveness factor versus Thiele modulus for a reaction which obeys a second-order rate expression. The catalyst is illuminated from the solution side.
Figure 5. Plots of effectiveness factor versus Thiele modulus for a reaction which obeys a Langmuir-Hinshelwood/Hougen-Watson rate expression. The catalyst is illuminated from the solution side. [KSCS0/(1 + KO20)] ) 1.0.
by eq 14. Upon substitution of eq 14 and the definition of the dimensionless rate
r* ) r/r0
(31)
into eq 7, the dimensionless form of the continuity equation for the substrate becomes
(1 + KO2CO20 + KSCS0) CS d 2C 2 ) Λ e-∆X (1 + KO2CO2 + KSCS) CS0 dX2
(32)
If one further assumes that oxygen is present in large excess and that its concentration is constant throughout the pore, it is possible to prepare plots of the effectiveness factor versus Thiele modulus for the case where the diffusional limitation is associated only with the solute molecules. This situation is that which is of greatest interest in our studies of photocatalytic remediation (see Sabate et al., 1991, 1992b). Figure 5 contains plots of the effectiveness factor versus the Thiele modulus for this situation. These plots have been prepared for a value of KSCS0/(1 + KO2CO20) equal to 1.0. While it is possible to choose values of this ratio such that eq 14 is effectively reduced to a first-order or a zeroorder model, we have found that the effectiveness factors are rather insensitive to large changes in this ratio. Comparison of Effectiveness Factor Plots. The effectiveness factor plots prepared for photocatalytic systems in which the catalyst slab is illuminated from the solution side (Figures 2-5) are qualitatively similar
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Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996
to the corresponding plots for conventional heterogeneous catalytic reactions which follow simple nth-order rate expressions (Carberry, 1976; Fogler, 1992; Hill, 1977; Satterfield, 1970). For cases in which the rate of the photocatalytic reaction is slow compared to the rate of the diffusion process (small Λ), the effectiveness factor is essentially independent of the Thiele modulus. When the reaction rate is rapid relative to diffusion, the effectiveness factor declines with increasing modulus. The influence of the optical film thickness on the effectiveness factors for photocatalytic systems is very significant. For a given optical film thickness, the maximum values of the effectiveness factors (i.e., the limiting values as Λ f 0) observed in systems which obey zero-, first-, or second-order kinetics, as well as LHHW rate expressions, are identical, but these limiting values can be much less than 1.0. The optical film thickness provides a dimensionless measure of the extent to which the incident radiation is attenuated as it passes through the photocatalytic membrane. For a specific rate expression, the value of the Thiele modulus at which the decline in the effectiveness factor from its low-modulus asymptote is observed increases as the optical film thickness increases. In other words, the relative importance of the role that the diffusion rate plays in the photocatalytic process decreases with increasing optical film thickness. The asymptotes observed at low Thiele moduli reflect the fact that even under circumstances for which reaction rates are slow relative to diffusion rates, the behavior of photocatalytic systems is quite different from that of conventional catalysts. Because of attenuation of radiation by the membrane in such situations and because of the associated effect on the reaction rate, the concentration profiles of all species within the pores will be relatively flat and the reaction rate expression can be approximated as
r ) r0e-µx or r ) r0e-∆X
(33)
regardless of the concentration dependence of the reaction rate. In these terms, eq 16 for the effectiveness factor then becomes
η≈
∫0Le-µx dx L
(34)
Upon evaluation of the integral, one finds for low Thiele moduli that
η)
1 - e-∆ ∆
for small Λ
Table 1. Comparison of the Values Obtained for the Effectiveness Factor from Equation 35 and the Orthogonal Collocation Solutionsa optical film thickness collocation solution asymptotic solution 1.000 3.000 7.000 10.000
0.631 90 0.316 70 0.142 70 0.100 00
a The tabular entries correspond to a first-order rate expression, illumination from the solution side, and a Thiele modulus of 0.05.
The plots in Figures 2-5 indicate the relative importance of the optical film thickness and the Thiele modulus in determining the value of the effectiveness factor for systems illuminated from the solution side. It is found that the effectiveness of the catalyst system is dominated by the Thiele modulus when the optical film thickness is small. For conventional systems (∆ ) 0), an order of magnitude increase in the Thiele modulus corresponds to a roughly 4-fold decrease in the effectiveness factor. However, at high optical film thicknesses, most of the photocatalyst is not exposed to light of significant intensity. The resultant shortage of photons retards the reaction rate and causes the concentration profile to be flat. Hence, the reaction rates in the interior of the catalyst are low, and the corresponding values of the effectiveness factor are small and relatively insensitive to changes in the Thiele modulus. Effectiveness Factors for Cases Where the Catalyst Is Illuminated from the Support Side The annular reactor geometry frequently used in studies of photocatalytic reactions employs a light source at the central axis of the reactor and a titania film on the inner wall of the annulus. In this configuration, the support side of the catalyst film is illuminated. This section summarizes the results of simulations of the photocatalytic behavior of such films. Recall that for this situation the boundary conditions on eq 7 are given by eqs 10 and 11. As in the previous discussion of the system illuminated from the solution side, we shall consider the problem of calculating effectiveness factors for zero-, first-, and second-order rate expressions, as well as LHHW kinetics. Zero-Order Kinetics. For zero-order rate expressions, a closed form solution to eq 19 can again be obtained for the concentration profile of the substrate. Integration of eq 19 subject to the boundary conditions of eqs 10 and 11 gives the following equation for the concentration profile
(35) C)1-
This asymptotic solution for the effectiveness factor at low moduli compares favorably with the results obtained via orthogonal collocation using nine collocation points (see Table 1). Inspection of the tabular entries indicates that eq 35 provides a good approximation to the lowmodulus asymptote. The high-modulus limits for effectiveness factors for conventional catalyst systems which obey simple nthorder rate expressions are of the form (Hill, 1977)
η ∝ 1/Λ
0.632 12 0.316 74 0.142 73 0.099 995
(36)
Plots of η versus Λ for photocatalytic systems are in agreement with this general result.
Λ2 [∆(1 - X) + e-∆ - e-∆X] ∆2
(37)
Since the concentration of any species cannot take on negative values, eq 37 is subject to the constraint that C g 0. Further physical constraints require that 0 e C e 1. Hence, eq 37 is applicable only when
Λ2 1 e 2 ∆ [∆ + e-∆ - 1]
(38)
This inequality serves to define a critical condition separating the regime in which the concentration of substrate is finite throughout the pore length from that in which the concentration of substrate goes to zero at
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 717
some critical (dimensionless) distance from the pore mouth, namely, Xc. For a fixed value of ∆, the critical value of the Thiele modulus, Λc2, at the boundary can be determined from inequality 38 as
Λc2 )
∆2 [∆ + e-∆ - 1]
(39)
For values of Λ e Λc, the concentration of reactant is finite throughout the pore length and eq 16 for the effectiveness factor becomes
∫01kφ0e-∆X dX ∆ 1 -∆X dX ) e ∫0 e η) 1 ∫0 kφ0e-∆ dX
(40) Figure 6. Plots of effectiveness factor versus Thiele modulus for a reaction which obeys a zero-order rate expression. The catalyst is illuminated from the support side.
Integration gives
η ) (e∆ - 1)/∆
(41)
For values of Λ in excess of the critical value, the concentration of substrate becomes zero at some critical value of X, namely, Xc. Recall that, for systems illuminated from the support side, X ) 1 at the pore mouth. For this situation the boundary condition given by eq 11 must be replaced by the following constraint
dC )0 dX
at X ) Xc
(42)
For this case, the concentration profile for Xc e X e 1 is given by
Λ2 C ) 1 - 2[∆e-∆Xc(1 - X) + e-∆ - e-∆X] ∆
(43)
while the expression for the effectiveness factor becomes
∫X1 e-∆X dX (e∆(1-X ) - 1) η) 1 ) ∆ ∫0 e-∆ dX c
c
(44)
A useful alternative form of eq 43 is
C)
[
X - Xc Λ2 -∆X + e - e-∆Xc + 1 - Xc ∆2 Xc - X -∆ (e - e-∆Xc) (45) 1 - Xc
(
)
]
The critical distance, Xc, can be determined using either eq 43 or eq 45. Equation 43 yields the relationship for Xc when one invokes the requirement that C must be zero at X ) Xc, and eq 45 gives the relationship for Xc when the second boundary condition at Xc is invoked; i.e., dC/dX ) 0 at X ) Xc. Thus both approaches indicate that
∆2 ) [e-∆ - e-∆Xc + ∆e-∆Xc(1 - Xc)] 2 Λ
(46)
Equation 46 may be solved numerically to determine Xc as a function of ∆ and Λ. These results are then substituted into eq 44 to generate the effectiveness factor plots shown in Figure 6. In order to obtain a convenient representation of the results for those cases where very large values of the effectiveness factor are
Figure 7. Plots of effectiveness factor versus Thiele modulus for a reaction which obeys a first-order rate expression. The catalyst is illuminated from the support side.
obtained, semilogarithmic coordinates are employed, not only in Figure 6 but also in the subsequent effectiveness factor plots for those cases in which the photocatalyst is illuminated from the support side. First- and Second-Order Kinetics. If the reaction of interest displays first-order or second-order kinetics, then eq 29 or eq 30 is applicable, and the appropriate boundary conditions are eqs 10 and 11. These problems can be solved using orthogonal collocation. The results can then be used in eq 16 to determine the effectiveness factor. The results of the numerical integration for first-order kinetics are shown in Figure 7, while those for secondorder kinetics are shown in Figure 8. LHHW Rate Expression. We have also calculated the effectiveness factors for the LHHW rate expression given in eq 14 and the boundary conditions of eqs 10 and 11. The appropriate differential equation is eq 32. If one assumes that the concentration of oxygen is constant throughout the pore, it is possible to prepare plots of the effectiveness factor versus Thiele modulus for the case in which the diffusional limitation is associated with the substrate molecules. Figure 9 contains plots of the effectiveness factor versus the Thiele modulus for this situation and a value of KSCS0/ (1 + KO2CO20) equal to 1.0. Plots are presented for several values of the optical film thickness, ∆. Comparison of Effectiveness Factor Plots. In terms of their dependence on Λ, the plots in Figures 6-9 are qualitatively similar to the corresponding curves for conventional heterogeneous catalysts. Note that all of these plots display a low-modulus asymptote that corresponds to the maximum effectiveness factor
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Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 Table 2. Comparison of the Values Obtained for the Effectiveness Factor from Equation 47 and from the Orthogonal Collocation Solutionsa optical film thickness collocation solution asymptotic solution 1.000 3.000 7.000 10.000
1.7178 6.3607 156.50 2202.4
1.7183 6.3618 156.52 2202.5
a The tabular entries correspond to a first-order rate expression, illumination from the support side, and a Thiele modulus of 0.05.
Figure 8. Plots of effectiveness factor versus Thiele modulus for a reaction which obeys a second-order rate expression. The catalyst is illuminated from the support side.
Figure 10. Plot of the effectiveness factor versus optical film thickness for a first-order rate expression with Thiele modulus as a parameter. The catalyst is illuminated from the support side.
Figure 9. Plots of effectiveness factor versus Thiele modulus for a reaction which obeys a Langmuir-Hinshelwood/Hougen-Watson rate expression. The catalyst is illuminated from the support side. KSCS0/(1 + KO2CO20) ) 1.0.
for a given optical film thickness. However, the magnitude of this limit is not bounded with respect to ∆. Note that for illumination from the support side of the photocatalyst, the effectiveness factors for optical film thicknesses greater than 1 are much greater than 1. This situation is analogous to the situation that arises in nonisothermal catalyst systems (Hill, 1977) where effectiveness factors much larger than 1 are often predicted. The origin of this situation lies in the defintion of r0 employed in eqs 6 and 7. We have defined r0 as the reaction rate at the surface of the catalyst closest to the reacting solution (X ) 0 for illumination from the catalyst side and X ) 1 for illumination from the support side). For systems that are illuminated from the support side of the catalyst, the maximum value of the local rate of reaction is not necessarily located at the interface between the catalyst and the reacting solution. For systems with a nonzero Thiele modulus and a nonzero optical film thickness, the local maximum in the rate may occur closer to the illuminated side of the catalyst than to the face exposed to the reacting solution. For a given optical film thickness, the location of the maximum in the rate in the catalyst slab moves closer to the solution side of the slab with increasing Thiele modulus. Systems that are illuminated from the support side are similar to conventional systems in the sense that the effectiveness factor for zero-order kinetics is insensitive to changes in Λ for Λ e Λc. Indeed, when the optical film thickness is equal to zero, the solution for the system illuminated from the support side is identical to the case for illumination from the solution side.
Furthermore, when ∆ ) 0, both of these solutions for the photocatalytic systems are identical to the solution for a conventional heterogeneous catalyst on which a zero-order reaction is occurring; i.e., eq 34 is applicable. The maximum values of the effectiveness factor for systems characterized by a finite optical film thickness and illumination from the support side of the catalyst film are largely determined by the value of the optical film thickness. The maxima are located in the region of low Thiele moduli. At low moduli, the concentration profile in the pore is relatively flat. Consequently, the local reaction rate can be approximated as r ) r0e-µx, and the limiting expression for the effectiveness factor in all cases becomes that obtained in eq 41 for zero-order kinetics. Hence, for small Λ
η ≈ (e∆ - 1)/∆
(47)
Values of the effectiveness factor calculated from both this asymptotic relation and the orthogonal collocation approach for first-order kinetics and Λ ) 0.05 are summarized in Table 2. Inspection of the tabular entries indicates that the asymptotic solutions compare very favorably with those obtained by orthogonal collocation using nine collocation points. Effectiveness factors for catalyst systems that are illuminated from the support side are strongly influenced by the value of the optical film thickness. Inspection of the plots in Figure 10 indicates that the effectiveness factor increases monotonically with optical film thickness. An order of magnitude increase in the optical film thickness (from 1 to 10) corresponds to an increase in the effectiveness factor of several orders of magnitude (typically from 1 to 2000). However, it should be emphasized that in this system a large effectiveness factor does not constitute optimal reaction conditions. The principal dilemma that occurs in photocatalytic systems that are illuminated from the support side is that both the resistance to radiation transfer and the
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diffusional limitations work together to limit the overall productivity of the system. In fact, it is possible to have extremely large effectiveness factors for photocatalytic systems that are illuminated from the support side while, at the same time, realizing only a small conversion of the reactants. Individuals interested in designing photocatalytic systems should attempt to avoid situations in which an optically dense catalyst is illuminated from the support side. In general, a full scale reactor should be constructed in a manner which maximizes the exposure of the catalyst to radiation. Although one might consider exposure of the catalyst to the radiation source a secondary consideration, our findings indicate that efficient transport of the radiation should be a primary design criterion. Conclusions The results of the model calculations given here should be useful in the design of photochemical reactors. For the configuration in which the illumination is introduced from the solution side of the catalyst film, the interpretation of the effectiveness factor variations follows closely that for conventional heterogeneous catalysis. Either diffusional limitations, at values of Λ generally greater than 1, or attenuation of the photon flux, at values of ∆ equal to or greater than unity, can reduce the global productivity of the catalyst to 10% of that of a very thin film with no concentration or photon flux variations. In the absence of diffusional limitations, the maximum value of the effectiveness factor corresponding to a specific value of ∆ may be determined from eq 35. The configuration with illumination coming from the support side of the film is less desirable with respect to enhancement of photocatalytic reaction rates but may be necessary, based on practical considerations associated with reactor construction. In this case, for the same incident flux of photons, the effective reaction-rate constant on the solution-side of the film is generally lower because of optical absorption in the film. On the other hand, substrate near the support side of the film will react with a higher effective rate constant. Direct comparison of the two configurations requires putting the two effectiveness factors on a common basis. The effectiveness factor calculated for support-side illumination is defined relative to the photon flux available at the solution side of the film, where it falls to its lowest level. Basing it instead on the photon flux on the support side of the catalyst introduces an adjustment factor of e-∆, the ratio of the photon fluxes at the front and rear of the film. With this correction, the effectiveness factors for both configurations behave in a qualitatively similar manner. In particular, the effects of the parameters Λ and ∆ on η are qualitatively the same. In the analysis of reaction rate data and in the design of a photocatalytic reactor for any particular reaction chemistry, the approach presented here should be used to account for nonuniform concentrations and photon-flux levels within the catalyst film. Nomenclature C ) dimensionless substrate concentration CO2 ) concentration of oxygen CS ) concentration of substrate species S in the fluid phase CS0 ) concentration of species S at the pore mouth D ) diffusion coefficient of species S k ) reaction rate constant
KO2 ) adsorption equilibrium constant for oxygen KS ) adsorption equilibrium constant for reactant S L ) average length of a pore LHHW ) Langmuir-Hinshelwood/Hougen-Watson n ) order of the reaction r ) reaction rate per unit surface area of the photocatalyst r0 ) reaction rate per unit area evaluated at the surface of the catalyst exposed to the fluid medium r* ) normalized reaction rate rj ) average pore radius x ) length coordinate parallel to the direction of the incident radiation X ) dimensionless distance Xc ) critical value of dimensionless distance Greek Symbols ∆ ) optical film thickness η ) effectiveness factor Λ ) generalized Thiele modulus Λc ) critical value of generalized Thiele modulus µ ) absorption coefficient per unit thickness of photocatalytic film φ ) flux of photons φ0 ) flux of photons at origin of distance coordinate
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Received for review May 10, 1995 Revised manuscript received November 13, 1995 Accepted November 23, 1995X IE950286U
X Abstract published in Advance ACS Abstracts, February 1, 1996.
