2D-LSPP: An Improved Line Source with Parallel Plane Emission

Ind. Eng. Chem. Res. 2007, 46, 7587-7597

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2D-LSPP: An Improved Line Source with Parallel Plane Emission (LSPP) Model for Photon Distribution in Annular Reactors Jose´ Pla´ cido and Claudio Augusto Oller Nascimento* Chemical Engineering Department, Polytechnic School of UniVersity of Sao Paulo, CEP 05508-900, Sa˜ o Paulo, SP Brazil

A new class of modelsstwo-dimensional line source with parallel plane emission, or 2D-LSPPshas been proposed for estimation of photon incidence in annular reactors. These models are similar to the well-known line source with parallel plane emission (LSPP) radial model, but these incorporate the axial distribution of photons at the boundary condition (i.e., at the reactor inner wall), which is a characteristic of the line source with spherical emission (LSSE) model. The new models have the advantage of easy implementation in numerical schemes, because only the radial direction must be considered during the calculations. Photon losses through the upper and lower cross-sectional cores are also automatically taken into account, as a function of the lamp length and the system geometry, which is a desirable model property in scale-up procedures that does not exist in the classical LSPP model. In this work, these new models are derived and simulation results are shown for comparison with the LSSE model. Some advantages of the new models are discussed, as well as their application limits. 1. Introduction The radial emission line source with parallel plane emission (LSSP) model, because of its simplicity and easy implementation, has been extensively used for the calculation of incident photon fields in annular photoreactors. However, this simplicity also imposes certain application limits. As an example, in freeradical polymerization activated by photochemical initiation,1 complex reactions may arise subsequent to the photon absorption process. Some of these may be highly dependent on the photon distribution shape, which is not considered in the LSPP model. For this reason, during the past three or four decades, models with increasing degrees of sophistication were developed, which consider the three-dimensional nature of the emissive process and, in some cases, the three-dimensional nature of the emitter lamp itself. A comprehensive review of these models is presented in the literature.2 Among these models, the following are cited: line source with spherical emission (LSSE), extense source with spherical surface emission (ESSE), and extense source with volumetric emission (ESVE). The first two models also have alternative versions for diffuse emission. The LSSE model is the simplest among these models;3 however, it can provide results that are quite similar to those from models that consider the three-dimensional nature of the lamp.4 Many works that have applied these radiation models are found in the literature.5-7 The use of such models presents several difficulties, because of factors such as (a) implementation complexity, because of their two- or three-dimensional nature, and (b) in most systems, the concentrations of the photon-absorbing species are not uniform, so there is a coupling between the photon field and the mass-transport equations. In this work, the LSPP and LSSE models are combined into a new model, which here is called the two-dimensional line source with parallel plane emission (or 2D-LSSP) model, which consists of an attempt to exploit the advantages of both models on which it is based. In the 2D-LSPP model, the LSPP unidimensional internal characteristic is preserved, and the axial * To whom correspondence should be addressed. Tel.; 55 11 3091 2216/1169. Fax: 55 11 3813 2380. E-mail address: [email protected].

distribution of the incident photons (which is a LSSE characteristic) is considered as the boundary condition. In contrast to the LSPP model, this new model can be applied to cases where the lamp length is smaller than the reactor length, representing a reasonable advantage. Moreover, its results are much closer to reality, because the axial distribution of the incident photons is taken into account. The total photon absorption values that are calculated from the LSPP and LSSE models generally are not the same. This is taken into account in the 2D-LSPP model by means of a correction factor. 2. Model Development The radial model is well-known in the literature; it is given by rad Grad λ (r) ) Gλ,o

()

Ro exp(-σλ,gδg) exp(r

r)r ∫r)R

in

σλ,c dr) (1)

where Grad λ,o is the average irradiance at the reactor inner wall for the radial model and is defined as rad ) Gλ,o

Sλ 2πRoL

(2)

The term in parentheses in eq 1 represents the transmittance of the inner wall (generally, a wall composed of Pyrex or quartz). The second exponential represents the photon transmittance in the annular space between Rin and r. If Beer’s law is valid, then nc

σλ,c )

κλ,jcj ) κλ,1c1 + κλ,2c2 + ‚‚‚ + κλ,n cn ∑ j)1 c

c

(3)

Note that if the concentration term appearing in eq 3 varies in the z-direction, then the incident photons in eq 1 are also a function of z. The LSSE model3 considers the lamp to be a succession of emissive points located at its axis. The photons that are emitted spherically, per unit time, by a point source placed on the lamp axis at height z′, are uniformly distributed over a spherical

10.1021/ie070078p CCC: $37.00 © 2007 American Chemical Society Published on Web 10/16/2007

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Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007

applied only to the interface where r ) Ro.] To calculate the photonic “irradiance” on the reactor wall at point P produced by all of the lamp emitter points, first, the (cylindrical) area element must be projected onto the photon beam direction. Physically, this means that the total photon rate must be preserved if either a spherical or cylindrical surface element is considered, i.e., s dGλ,in dA ) dGλ,in dAs ) s s dA cos θ ∴ dGλ,in ) dGλ,in cos θ (8) dGλ,in

Therefore,

dGλ,in ) cos θ )

dGsλ )

Sl,λ dz′ 4πF

2

(Fr)] exp[-σ

[

exp -σλ,gδg

λ,c(r

rad Grad λ (r) ) Gλ,in

( )

Rin exp (r

r)r ∫r)R

in

σλ,c dr)

(Fr)] (4)

Gλ,in(z) )

Sl,λ dz′ 4πF

2

( )

Sλ rad F ) Gλ,in Ftz (10) 2πRinL tz

z′)l +l dGλ,in ) ∫z′)l o

o

Ftz ) Ftz(ηl ,ηlo,ηL,ηz) )

[x

1 2(ηl /ηL)

ηlo + ηl - ηz

-

1 + (ηlo + ηl - ηz)2

]

ηlo - ηz

x

(11)

1 + (ηlo - ηz)2

The aVerage “irradiance” over the entire inner surface is obtained by integration of eq 10 between z ) 0 and z ) L, the result of which is given by

(5) G h λ,in )

1 ηL

∫ηη)0)η z

z

L

Gλ,in (ηz) dηz ) rad Gλ,in

(6)

( )∫ 1 ηL

ηz)ηL

ηz)0

rad Ftz dηz ) Gλ,in FT (12)

where FT is called the “View factor”,

Another important restriction is related to the reactor nominal boundaries to be considered during the derivations. Figure 1 presents three distinct regions a, b, and c, with axial lengths La, L, and Lc, respectively. Regions a and c exist when the entrance and exit sections are far away from the nominal boundaries (i.e., from z ) 0 and z ) L). In the present work, only the nominal reaction section of the reactor will be considered (i.e., only the region b in Figure 1). Considering σλ,g ) 0, the incident photons at r ) Rin, from eq 4, show that the (differential) “irradiance” on a spherical surface element located at point P, produced by an emitter point placed at z′, is given by the following equation: s ) dGλ,in

2

where Ftz is a view factor that is conveniently expressed in terms of the reduced variables η. Thus,

where

Sλ rad ) Gλ,in 2πRinL

(9)

x ( ) z′- z Rin

Equation 9 is then integrated along the lamp axis, to obtain the “irradiance” on a cylindrical surface element placed on the inner wall at height z, which is given by

- Rin)

This model will be used to derive relations concerning the incident photon rate at the reactor inner wall. The main hypotheses adopted in the models presented in this work are (i) there is no radiation emission inside the (homogeneous) photoreactor, (ii) reflection and refraction effects are negligible, and (iii) there is no scattering of photons. In this work, we will restrict the analyses to reactors that present no absorbing inner glass wall (i.e., σλ,g ) 0). This is valid, for example, in the case of quartz walls for usual spectral bands. For this case, eq 1 may be rewritten as

1

1+

Figure 1. Geometry of the reactor and photon emission pattern. (After Romero et al.4)

surface of radius Fo, centered at the emission point, as shown in Figure 1. This is the boundary condition from which the model was derived. The resulting model equation is given by

Sl,λ dz′ cos θ 4πF2

(7)

[Note that, here, we use the word “irradiance” with quotation marks to mean the photon rate per unit surface, which is incident on r ) Rin, for short. However, the term irradiance is generally

FT )

x

1 [ 1 + (ηlo - ηL)2 - x1 + ηlo2 2ηl

x1 + (η +η - η ) + x1 + (η + η ) ] (13) 2

lo

l

2

lo

L

l

which is valid for line sources with a spherical emission pattern. For lamps placed at the middle of the reactor, i.e., when lo ) (L - l)/2, it follows that

FT )

1 ηl

[x ( ) x ( ) ] 1+

ηL + ηl 2 2

1+

ηL - η l 2

2

(14)

It can be shown that eq 14 converges to the correct limit for some limiting cases, such as

lim FT ) 1 Lf∞

and

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7589

lim FT ) 1 Rinf0

For the case where l ) L (lo ) 0), it follows that

FT )

1 ( 1 + ηL2 - 1) ηL x

(15)

Equation 15 constitutes a particular case of the general expression given by eq 13.8 The view factor could also have been obtained by the following definition:

photon rate received at (or incident on) the reactor inner wall surface FT ) total rate of photon emission

Note that Gsλ,o must not be called “irradiance”, because, during its derivation, the area element was not projected onto the photon beam direction. For this reason, there is no direct comparison between this term and Gλ,in shown in eq 10. The s is based is smaller than that on which Gλ,in area on which Gλ,in is based, and the consequence is that the average integration of Ftsz along direction z does not give the correct value of the view factor:

FsT )

1 ηL

(16) 0.5 ln

or

FT )

2πRinLG h λ,in h λ,in G ) rad Sλ G

(17)

∫ηη)0)η z

[

z

L

Ftsz dηz )

1 + (ηL - ηlo) 1 + ηlo2

]

{ [

2

- (ηL- ηlo - ηl) arctan(ηL - ηlo -

ηl) +(ηlo + ηl) arctan(ηlo + ηl) + (ηL- ηlo) arctan(ηL -

λ,in

which is equal to eq 12. As can be seen in eq 13, the view factor is dependent on the ratio of the reactor length, the lamp length, and the lamp position to the reactor inner radius, Rin. For scale-up concerns, all these ratios should be kept constant, to keep the same photon losses (1 - FT) in both systems. Furthermore, this also would preserve the relative incident photon distribution, as can be seen in eq 11. The idea for the derivation of the new model (the 2D-LSPP1 model) is quite simple: the local “irradiance”, given by eq 10, is substituted into eq 5 for the radial model average “irradiance” at r ) Rin. The new model then is given by

Gλ(r,z) )

SE/PP Pabs,λ Ftz(z)Grad λ (r)

SE/PP (Pabs,λ Sλ ) Ftz exp(2πrL

∫Rr

in

σλ,c dr) (18)

SE/PP , has been introduced. This where a correction factor, Pabs,λ factor represents the ratio between the average absorbed photon rate (inside the entire reactor) calculated by the LSSE model and that calculated by the LSPP model, based on the incident photons (i.e., the LSPP model corrected by the view factor). An approximate expression will be derived for this correction factor in a separate section. Basically, the objective of 2D-LSPP class models is to obtain an approximation for the distribution of incident photons inside the photoreactor. The first attempt to do this, preserving the total rate of photons arriving at the reactor inner wall, led to the 2D-LSPP1 model. However, this does not mean that the derived model exactly represents the existing situation inside the photoreactor. It can be shown that, in the LSSE model, the incident photon rate per unit area at a point P located on the photoreactor inner wall at height z is given by

s (z) ) Gλ,in

S

z′)l +l λ s rad s dGλ,in ) Ft Fs ) Gλ,in ∫z′)l 2πRinL t o

o

z

z

(19)

where

Ftsz )

ηlo) - ηlo arctan(ηlo)

1 [arctan(ηz - ηlo) - arctan(ηz - ηlo- ηl)] (20) (2ηl/ηL)

This expression gives the distribution shape of the photons arriving at the inner wall. A simplified version of eq 20 was presented elsewhere3 (using ηlo ) 0, ηl ) ηL).

}

(21)

To put the profiles shown in eqs 10 and 19 on a same basis, and make them comparable, the factor in eq 19 (FsT) may be normalized by its average along the photoreactor inner wall and multiplied by the view factor. Hence,

G′λ,in )

( )

Sλ FT s rad F ) Gλ,in F′tz 2πRinL Fs tz T

where

F′tz )

)

]

1 + (ηL- ηlo - ηl)2 1 0.5 ln 2ηl 1 + (ηlo + ηl)2

(22)

() FT

FsT

Ftsz

Therefore, a new model version may be obtained if the factor F′tz is substituted for Ftz in eq 18. The new model is then given by SE/PP Gλ(r,z) ) Pabs,λ F′tz(z)Grad λ (r) )

F′tz

SE/PP (Pabs,λ Sλ) exp(2πrL

r)r ∫r)R

in

σλ,c dr) (23)

Any linear combination between the factors from these two models could also represent an alternative model factor for the photon distribution at the photoreactor inner wall, provided that the total photon incidence along the inner wall is preserved (the linear combination weights must amount to 1). The factors Ftsz, Ftz, F′tz, and their average integration along the reactor height, are shown in Figures 2a and 2b for a photoreactor with a length (L) of 20 cm and inner radius (Rin) of 2 cm. In Figure 2a, an L ) 4 cm lamp is used; in Figure 2b, an L ) 14 cm lamp is used. For these lamp/reactor geometries, the view factors are 98% and 96.5%, respectively. Despite greater photon losses, relatively larger lamps produce more uniform photon fields, as can be observed in the figures. The photon incidence calculated by the factor Ftz is greater at the central axial region, and smaller at the ends, than that calculated by the factor F′tz. It is possible to compensate the greater photon loss in the case shown in Figure 2b by changing the lamp strength. With regard to the systems depicted in Figures 2a and 2b, if Rin ) 4 cm instead of Rin ) 2 cm, the view factors would be equal to 92.6% and 89%, respectively. Therefore, it is important that these different photon losses be properly considered in

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the system (κλ,i ) 0, i * j). Notice that concentrations in this equation can also vary along the z-direction. For uniform concentrations inside the entire reactor (a situation approached, for example, in a reactor/tank with recycling system with a high recycle flow rate value), eq 26 may be integrated along the z-direction to obtain V SE/PP VRPabs,λ,j ) FTPabs,λ Sλ

[ ]

κλ,jcj [1 - exp(-τλ,c)] σλ,c

(27)

where τλ,c is known as the optical density:

τλ,c ) σλ,c(Rout - Rin)

(28)

If we had used the radial model for this derivation, we would have obtained the following expression: PP ) Sλ VRPabs,λ,j

Figure 2. Photon distribution at the reactor inner wall for a system where Rin ) 2 cm and ηL ) 10: (a) ηl ) 2, ηlo ) 4, FT ) 0.9798 and (b) ηl ) 7, ηlo ) 1.5, FT ) 0.9651.

reactor scale-up procedures. Ideally, the view factor itself should be preserved in these procedures, which may be done if ηL, ηl, and ηlo are all kept constant. The 2D-LSPP1 model will be used to derive the local and average (spectral) volumetric rate of photon absorption (LVRPA); however, the resulting expressions are also valid for the 2D-LSP2 model, provided that the factor F′tz is substituted for Ftz. With the assumption that Beer’s law is valid, then, from the 2D-LSPP1 model (eq 18), the spectral LVRPA of the V , is obtained as j-component, which, here, is called Pabs,λ,j V Pabs,λ,j (r,z) ) κλ,jcjGλ(r,z) ) SE/PP Pabs,λ Sλκλ,jcj Ftz exp(2πrL

r)r ∫r)R

in

σλ,c dr) (24)

The spectral LVRPA, averaged over the cross section, is then given by V (z) ) P h abs,λ,j

1 (VR/L)

r)R ∫r)R

out

in

V Pabs,λ,j (r,z)2πr dr

V P h abs,λ,j

SE/PP Pabs,λ Sλκλ,jcj (z) ) Ftz VR

r)R ∫r)R

out

in

exp[-σλ,c(r - Rin) dr]

κλ,jcj [1 - exp(-τλ,c)] σλ,c

(29)

SE/PP ) is dependent on both Generally, the correction factor (Pabs,λ system geometry and average concentration inside the photoreactor, as well as the spectral characteristics of the absorbing species. It also is dependent on the concentration profiles of all the absorbing species. For polychromatic systems, where various absorbing species exist, it can be also considered to be independent of the wavelength and calculated as a global factor, which would be valid as an average correction for the given concentration and spectral characteristics of the system. Despite all these dependences, for low concentrations, the error of the average factor is small. Thus, this factor may be calculated by taking the inlet stream concentration as a basis and keeping it constant in subsequent calculations or, if desired, it may be continuously updated to consider the concentration changes. For a given geometry, its calculation can be performed a priori, the results being stored as a function of the inlet stream concentration. It may be obtained also based on actinometry experiments. An empirical approximation for this factor is proposed for a system that consists of a photon source enclosed by a nonabsorbing glass wall (σλ,g ) 0; e.g., quartz walls) and uniform concentrations inside the annular space. With the aforementioned hypotheses, the correction factor is defined by the following expression:

SE/PP Pabs,λ =

(25)

If the concentrations are uniform along the radial direction at height z, an analytical expression then follows for this average LVRPA:

[ ]

SE Pabs,λ PP FTPabs,λ

)

SE VRPabs,λ

SλFT[1 - exp(-τλ,c)]

(30)

where eq 29 was used for the total amount of photons absorbed PP . by all components in the spectral range λ to obtain Pabs,λ (Beer’s law, eq 3, is not necessary for this derivation.) To derive an expression for this factor, the limiting case as the optical density approaches zero is analyzed first. From eq 4, neglecting the inner wall absorption and taking the limit as σλ,c approaches zero, one obtains

or V (z) ) P h abs,λ,j

[ ]

SE/PP Pabs,λ Sλ

κλ,jcj

VR

σλ,c

Ftz

{1 - exp[-σλ,c(Rout - Rin)]} (26)

The term between brackets in eq 26 (κλ,jcj/σλ,c) is equal to 1 when the component j is the only photon-absorbing species in

dGsλ )

F exp -σλ,c(r - Rin) = r 4πF2 Sl,λ dz′ F 1 - σλ,c(r - Rin) r 4πF2

Sl,λ dz′

[ [

]

]

(for σλ,c f 0) (31)

‘This expression may be applied in the definition for the total amount of absorbed photons, given by

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7591 SE VRPabs,λ )

R L v 2πrPabs,λ dr dz ) ∫z)0 ∫r)R l +l L R 2πr[σλ,c ∫z′)l dGsλ] dr dz ∫z)0 ∫r)R out

in

out

o

in

(32)

o

An analytical solution can be obtained for eq 32; only the final result is given here. After dividing this result by expression 33,

SλFT[1 - exp(-τλ,c)] = Sλτλ,cFT ) Sλσλ,cRin(ηR - 1)FT

where

F)

[

2

a22 ln

ηR2 ln

]

[ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )]

1 a 2 ln 2ηl 12 a122 + 1

a222 + ηR2 a222 + 1

a122 + ηR2

a11 + ηR 2

2

- ln

ηR2 ln

- a112 ln

+

a212

ln

a122 + 1

a112 + ηR2 a112 + 1

a212 + ηR2 a212 + 1

- ηR2 ln

a112 + 1 a222 + ηR2 a212 + ηR2

coj

)1-

1 ln{1 + [exp(τλ,j)- 1] exp(-ktr)} τλ,j

(

+ ln

τλ,j ) κλ,jcoj (Rout - Rin)

( )[

k ) φλ,j

v Pabs,λ,j )

+

a112 + 1 a222 + 1 a212 + 1

-

(35)

a11 ) ηlo; a12 ) ηlo - ηL; a21 ) ηlo + ηl ; a22 ) ηlo + ηl - ηL (36)

coj

VR

FT

(41)

In eq 34, the factor is obtained from an equation that is similar to eq 21, except that, in this case, the outer radius Rout, instead of the radius Rin, must be used in the definitions of ηl , ηlo, and ηL (e.g., ηl ) l /Rout, etc). The second step is to analyze the limiting case as the optical density approaches infinity. Because all incident photons are SE ) SλFT. Hence, from eq 30, absorbed in this case, then VRPabs,λ SE/PP it can be observed that Pabs,λ ) 1 as τλ,cf ∞. From these two limiting cases, the following empirical expression is proposed as an approximation for the correction factor, which is valid when σλ,g ) 0: SE/PP SE/PP SE/PP Pabs,λ ) Pabs,τ + (1 - Pabs,τ )[1 - exp(-τλ,c)] (37) λ,cf0 λ,cf0

Equation 37 provides exact values in the two limiting situations SE/PP SE/PP analyzed: (1) in the limit as τλ,c f 0 (Pabs,λ ) Pabs,τ ) and l,cf0 SE/PP (2) in the limit as τλ,c f ∞ (Pabs,λ ) 1). For intermediate values of τλ,c, the error involved in expression 37 will be dependent on the source/system geometry. These errors are analyzed in Appendix A for several test cases. 3. Application to the Decomposition of Species j at Steady State In several situations, for monochromatic lamps, decomposition of the photon-absorbing species at steady state may be obtained by an equation of the type

(38)

1 L

∫z)0

z)L

v P h abs,λ,j dz )

coj xj trφλ,j

(42)

These results are also valid for the 2D-LSPP2 model. Equation 39 is similar to the relations obtained for tubular photoreactors.9 For polychromatic lamps, the term in parentheses SE/PP Sλ, may be replaced by (PSE/PP in eq 41, Pabs,λ abs S) (the total source strength S emitted in spectral ranges where species j absorbs photons, corrected by an average, nonspectral correction factor), and an average absorption coefficient may be used in these equations instead of κλ,j, which would be given by

κavg,j )

{

s FT,R out

dcjj v )R h j ) - φλ,jP h abs,λ,j (z) dz

(40)

]

τλ,j

SE/PP (Pabs,λ Sλ)

For this case, eq 26 may be integrated along z to obtain the aVerage spectral Volumetric rate of photon absorption for the entire reactor, given by

-

a122 + 1

)

L (39) Vjz

where

(34)

and

vjz

jcLj

(for τλ,c f 0) (33)

1 1 η Fs - FsT + F 4 (ηR - 1)FT R T,Rout

a122 + ηR2

xj ) 1 -

for tr )

and neglecting all terms that contain σλ,c, the solution can be represented by eqs 34-36: SE/PP Pabs,τ ) λ,cf0

When the jth component is the only photon-absorbing species v (z) from eq 26, and Beer’s law applies, after substituting P h abs,λ,j eq 38 may be integrated from z ) 0 to z ) L to obtain

-1 cjj(Rout - Rin) nλ

ln 1 -

∑

k)1

×

()

SE/PP Sλk Pabs,λ k (1 - exp[-κλk,jcjj(Rout - Rin)]) S PSE/PP abs

}

(43)

where cjj is an appropriate average concentration. For small conversions, cjj ) coj may be used as an approximation. In this equation, the quantum yield was considered to be wavelengthindependent. This expression was derived from the relations that follow. For polychromatic lamps, the total decomposition rate of the component j is obtained as a summation over all the relevant spectral contributions, i.e., ηλ

-Rj )

V ∑ φλ ,jPh abs,λ ,j )

k)1

Ftz

k

k

ηλ

∑

VR k)1

SE/PP φλk,jSλkPabs,λ k

κλk,jcjj σλk,c

{1 - exp[-σλk,c(Rout - Rin)]} (44)

If a monochromatic-like equation is used and the quantum yield is wavelength-independent, then the aforementioned equation for the (FT,Rout) case may be written as

-VRRj FtzφjSPSE/PP abs

)

nλ

SE/PP Sλk Pabs,λ k

k)1

S PSE/PP abs

∑

{1 - exp[-κλk,jcjj(Rout - Rin)] )

1 - exp[-κavg,jcjj(Rout - Rin)]} (45)

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Equation 43 follows directly from eq 45. The spectral distribution of the lamp output may be assumed to be equal to that provided in the manufacturer’s data (i.e., Sλk ) STotal(Fλk/F)). In this case, STotal also could be evaluated using some experimental method (e.g., actinometry). The average molar absorption coefficient is dependent on the concentration but may be kept constant in the calculations as an approximation. Otherwise, no analytical solution is possible and a numerical procedure must be used to solve the system given by eqs 38 and 44. 4. Comparison between the 2D-LSPP1, 2D-LSPP2, and LSSE Models To compare these models, the conditions shown in the caption of Figure 2 were used, with Rout ) 4 cm. The same monochromatic source strength was used in all cases. The rate of photon absorption was calculated and normalized as follows: norm Pabs,λ,j )

v VRPabs,λ,j Sλ

(46)

To compare the models under different levels of photon absorption, four optical density values were used: τλ,j ) 0.02, 0.20, 2.0, and 20.0. Results are shown in Figures 3 and 4 for l ) 14 cm and in Figures 5 and 6 for l ) 4 cm. In these figures, results from the LSSE model have different symbols, according to the radial position inside the photoreactor. Full lines represent norm values always deresults from the 2D-LSPP models. Pabs,λ,j crease from the inner wall (radial position ) 0.00) to any position inside the photoreactor. Because of axial symmetry, the right-hand side of these figures is shown in semilogarithmic scale, to facilitate the visualization of the results. Figures 3 and 4 show that, for systems where l/L is large (0.7 in these figures), both the 2D-LSPP1 and 2D-LSPP2 models give similar results. For dilute conditions (τλ,j up to ∼0.2), the 2D-LSPP2 model gives better results at the reactor borders, whereas at intermediate τλ,j values (τλ,j = 2), the 2D-LSPP1 model seems to be more precise. However, care must be taken if these models are to be used for optical densities greater than τλ,j ≈ 2, because, depending on the system, large errors may result. This can be observed in Figures 5 and 6, where a relatively small l/L value (0.2) was used. The 2D-LSPP2 model again gives better results for dilute conditions up to τλ,j ) 0.2. No reasonable results were obtained from either model for points next to the reactor borders at optical densities of g2. For points near the middle of the reactor, however, the 2D-LSPP1 model seems to give better results than the 2D-LSPP2 model for these higher optical densities. This may be explained by the fact that, in the 2D-LSPP1 model, the incident photons at r ) Rin are more concentrated at the middle of the reactor. This is more similar to the LSSE model at high values of τλ,j. For small values of τλ,j, excessive photon incidence occurs at the middle of the reactor using the 2D-LSPP1 model, SE/PP because the correction factor is >1 (e.g., at τλ,j ) 0.02, Pabs,λ k ) 1.343). In the 2D-LSPP2 model, the factor Ftsz, which gives the exact photon distribution of the LSSE model at r ) Rin, must be divided by the factor FsT/FT, to preserve the photon balance SE/PP , to correct for in the system, and then be multiplied by Pabs,λ k the total number of absorbed photons in the annular space. The factor FsT/FT is dependent only on the system geometry and is given in the captions of the corresponding figures. For dilute conditions, both these factors have similar values. Therefore, at low optical densities, the 2D-LSPP2 model is more similar to the LSSE model.

Figure 3. Dimensionless absorbed photon distribution. Comparison between the LSSE model (points) and the 2D-LSPP1 models (lines); l ) 14 cm. Radial position ) (r - Rin)/(Rout - Rin).

5. General Comments The 2D-LSPP models were used to study methyl methacrylate (MMA) free-radical polymerization with photochemical initiation.1 In this system, the resulting polymer (PMMA) polydispersity was quite sensitive to the photon distribution inside the annular space. The monomer conversion was also dependent on this distribution. The results from the 2D-LSPP and LSSE models are comparable for these variables, provided that the SE/PP correction factor Pabs,λ are properly taken into consideration.


Figure 4. Dimensionless absorbed photon distribution. Comparison between the LSSE model (points) and the 2D-LSPP2 models (lines); l ) 14 cm. Radial position ) (r - Rin)/(Rout - Rin). FsT/FT ) 1.380.

Figure 5. Dimensionless absorbed photon distribution. Comparisons between the LSSE model (points) and the 2D-LSPP1 models (lines); l ) 4 cm. Radial position ) (r - Rin)/(Rout - Rin).

In systems where the inner wall is photon-absorbing (σλ,g * 0), both the view factor and the related distribution factors could be computed at the inner surface (i.e., at r ) Ro ) Rin - δg, and the 2D-LSPP models would then give the photon distribution at this surface. In this case, the glass-transmittance term, exp(-σλ,gδg), should be considered (as a multiplying factor) in the definition of both models. As a consequence, a numerical SE/PP (the definition computation of the correction factor, Pabs,λ k given in eq 30), will be necessary.

Finally, in actinometry experiments, the estimated lamp strength is an effectiVe Value, that is dependent on the model chosen for the photon field calculation. Using the 2D-LSPP model, it is given by SE/PP Sλ,eff ) Pabs,λ SλFT

(47)

in this case, a closer estimation of the real value of Sλ would result, independent of the reactor geometry, which is taken into

7594


recommendations presented here may be used with caution, to serve as rough guidelines for application of the proposed equations. In practice, the user should test both models and verify the one that gives the best results. The final equation for the 2D-LSPP1 model is given by eq 18:

Sλ SE/PP Gλ(r,z) ) Pabs,λ Ftz exp(2πrL

r)r ∫r)R

in

σλ,c dr)

(48)

where Ftz is given by eq 11. This model is used for intermediate optical densities (τλ = 2). This equation may also be used for higher optical densities in cases where the photon profile near the reactor ends (reactor inlet and outlet) has little influence on the desired results. The final equation for the 2D-LSPP2 model is given by the equation SE/PP F′tz Gλ(r,z) ) Pabs,λ

Sλ exp(2πrL

∫Rr

in

σλ,c dr)

(49)

where F′tz is taken from the definitions given by eqs 20-22 and eq 13. This model is used for low optical densities. SE/PP is calculated by In eqs 48 and 49, the correction factor Pabs,λ the approximate expression given by eq 37 and its subsidiary definitions (eqs 34-36), which are valid when σλ,g ) 0. The local (spectral) volumetric rate of photon absorption (LVRPA) is given by eq 24: v (r,z) ) κλ,jcjGλ(r,z) Pabs,λ,j

(50)

where the incident photons may be taken from either eq 48 or eq 49. For uniform concentrations along the radial direction, it is possible to obtain an expression for the cross-sectional average of the (spectral) volumetric rate of photon absorption, which is given by eq 26, based on the 2D-LSPP1 model,

Sλ κλ,jcj v SE/PP Pabs,λ,j (z) ) Pabs,λ Ftz {1 - exp[-σλ,c(Rout - Rin)]} (51) VR σλ,c or substitute F′tz for Ftz in this equation to switch from the 2DLSPP1 model to the 2D-LSPP2 model instead. For perfectly mixed annular reactors, it is possible to obtain an expression for the average (spectral) volumetric rate of photon absorption that is valid for the entire reactor, which is given by

Sλ κλ,jcjj v SE/PP Pabs,λ,j ) Pabs,λ FT {1 - exp[-σλ,c(Rout - Rin)]} (52) VR σλ,c

Figure 6. Dimensionless absorbed photon distribution. Comparison between the LSSE model (points) and the 2D-LSPP2 models (lines); l ) 4 cm. Radial position ) (r - Rin)/(Rout - Rin). FsT/FT ) 1.399. SE/PP account by the factors FT and Pabs,λ . Using the LSPP model, the estimated Sλ will be geometry-dependent. Then, a new actinometry experiment should be done to get the proper Sλ value for any new geometric configuration. This is an advantage of using 2D-LSPP models versus the LSPP model.

If several components absorb photons, then the correction factor SE/PP Pabs,λ must be obtained for the extinction coefficient of all components (σλ,c), which may be calculated using eq 3 when Beer’s law is applicable. The fraction of photons that do not impinge on the inner wall will probably be lost in reactors whose boundaries are welldefined. This fraction is equal to 1 - FT, where the view factor is given by eq 13:

FT )

x

1 [ 1 + (ηlo - ηL)2 2ηl

x1 + η

2 lo

-

x1 + (η + η - η ) + x1 + (η 2

6. Design Equations In this section, a summary of the main results from this work, together with appropriate application limits, is provided. The

lo

l

L

lo

- ηl )2] (53)

The fraction of photons that are directed into these annular surfaces represents a potential risk of loss, which is maximum


Figure A1. Predicted correction factor (from eq 37) versus optical density for several source/system geometries. Curve numbers refer to the experiments shown in Table A1.

Figure A2. Relative percentage error in the predicted correction factor versus optical density for several source/system geometries. Curve numbers refer to the experiments shown in Table A1.

in the limit as the optical density approaches zero. This loss would be equal to zero in a system where the optical density is infinite, and intermediate losses are expected for optical densities between these limits.

Potential fraction of photon loss ) FT - FT,Rout

(54)

7. Conclusions A new class of models is proposed for estimation of photon incidence in annular reactors. These models are similar to the well-known radial model (line source with parallel plane emission, LSPP), but the axial distribution of photons at the boundary condition (i.e., at the reactor inner wall) is based on the line source with spherical emission (LSSE) model. The new models have the advantage of being easily implemented in numerical schemes for solution, together with the transport equations, because only the radial direction must be considered during the calculations. Photon losses through the upper and lower cross-sectional cores are taken into account in these new models, as a function of the lamp length and the system geometry, which is a desirable model property to be considered during any scale-up procedure. This characteristic

does not exist in the classical LSPP model, where the source and reactor lengths must be the same. Simulation results indicate that the 2D-LSPP2 model gives the best results at optical densities up to ∼0.2, whereas the 2DLSPP1 model provides better results at intermediate optical densities (e.g., at ∼2). These models are good starting points for the estimation of the photon incidence in annular reactors, with the advantage of providing analytical expressions for comparison and testing of numerical schemes. Because of their simplicity, they are ideal for implementation of photon field calculations in computational fluid dynamics (CFD) codes. Because of the high complexity of some systems, this may be a great advantage over moresophisticated and more computationally demanding photon field models. Appendix. Testing the Proposed Empirical Expression for SE/PP Pabs,λ To verify the errors involved in the application of eq 37, a full four-level (24) “simulation” design was performed as follows.

7596


Table A1. Numerical Simulation Results for Testing eq 37 design 0000 ++++ +++++-+ ++-+-++ +-++--+ +---+++ -++-+-+ -+---++ --+---+ ----

centerb sim1 sim2 sim3 sim4 sim5 sim6 sim7 sim8 sim9 sim10 sim11 sim12 sim13 sim14 sim15 sim16

FT

FT,Rout

FsT

0.8903 0.7808 0.6180 0.8937 0.7036 0.7808 0.6180 0.8937 0.7036 0.9512 0.9050 0.9950 0.9798 0.9512 0.9050 0.9950 0.9798

0.7952 0.6770 0.4806 0.7799 0.5279 0.7621 0.5913 0.8754 0.6693 0.8198 0.6770 0.9279 0.7771 0.9278 0.8612 0.9888 0.9563

1.1315 0.9717 0.7048 1.1061 0.7821 0.9717 0.7048 1.1061 0.7821 1.3710 1.2404 1.4708 1.3709 1.3710 1.2404 1.4708 1.3709

SE/PP Pabs,τ λf0

1.1639 1.1240 0.9772 1.1142 0.9362 1.2223 1.1089 1.2155 1.0774 1.2544 1.0900 1.3334 1.1355 1.4038 1.3109 1.4533 1.3502

a

Err% max τλ 0.88 1.65 4.61 0.89 5.69 -0.94 -0.81 -1.61 -0.68 1.51 3.65 -0.96 -0.92 2.27 0.68 2.37 -0.56

4.0 3.4 2.4 4.2 2.2 1.2 0.8 1.0 0.8 2.6 2.8 1.4 0.8 0.4 0.4 0.6 1.8

Note that the inner wall absorption was neglected in these derivations. Therefore, these results must not be used for systems where such an assumption is not applicable. Acknowledgment The authors would like to thank CNPq, CAPES/DAAD, and FAPESP for financial support and fellowships. Notation cj

molar concentration of component j [mol/m3]

cjj

cross-sectional average molar concentration of component j [mol/m3]

dA

surface element area [m2]

dGλ,in

“irradiance” at r ) Rin due to a point source [mol s-1 m-2]

dqF

F-component of the radiative flux vector in spherical coordinates [mol s-1 m-2]

Ftz

lamp axis-to-differential cylindrical surface element view factor, eq 11

Fstz

lamp axis-to-differential surface element factor, eq 20

F′tz

normalized axis-to-differential surface element factor, eq 22

FT

lamp axis-to-inner wall cylindrical surface view factor, eq 13

FT,Rout

lamp axis-to-outer wall cylindrical surface view factor (similar to eq 13, except that Rout must be used instead of Rin in the dimensionless variables η)

FsT

lamp axis-to-surface average factor at r ) Rin, eq 21

s FT,R out

lamp axis-to-surface average factor at r ) Rout (similar to eq 21, except that Rout must be used instead of Rin in the dimensionless variables η)

Fλk/F

photon source spectral distribution (from the manufacturer’s data)

Gλ

incident photon rate per unit area [mol s-1 m-2]

Gλ,in

“irradiance” on a cylindrical surface element at r ) Rin [mol s-1 m-2]

rad Gλ,o

average irradiance on a surface element at the reactor inner wall taken from the

LSPP

(radial) model, eq 2 [mol s-1 m-2]

s Gλ,in

incident photon rate per unit area at r ) Rin [mol s-1 m-2]

I

light intensity [mol s-1 sr-1 m-2]

l

lamp length [m]

lo

lamp position [m]

L

reactor length [m]

nc

total number of components

nλ

total number of relevant spectral ranges (where there is significant photon absorption by the absorbing species)

v Pabs,λ,j

local spectral volumetric rate of photon absorption of component j [mol s-1 m-3]

a

τλ (( 0.2), in this table, is the optical density where the percentage error is maximum. b A value of l ) (4 + 20 + 40)/3 = 21 cm was adopted as the center for the source length.

(1) The following design parameters were selected: Rin, ∆R ) Rout - Rin, l, and L. The photon source was placed at the center (lo ) (L - l )/2). (2) The following ranges were considered: for Rin, 2-10 cm; for ∆R, 1-6 cm; for l, 4 cm to 100% of L f l ) L); and for L, 20-40 cm. These values give the following dimensionless ranges: ηR - 1 ) 0.1-3.0, ηl ) 0.4-20, and ηL ) 2-20. For each experiment, predictions from eq 37 were compared to numerical computations for optical densities varying from 0.002 to 12 at increments of 0.2 (generally). The numerical results are shown in Figure A1, and the corresponding relative percentage errors are shown in Figure A2. From these figures, the maximum relative percentage error and the corresponding optical density were obtained for each experiment. These data are shown in the two last columns of Table A1. In this table, the view factor FT (calculated at both r ) Rin and r ) Rout), the factor FsT, and the correction factor SE/PP ) are also presented. for dilute conditions (Pabs,λf0 The maximum errors generally were 30%, calculated as 100(1 - FT)), and (b) potential photon losses through both the lower and the upper cross-sectional areas of the reactor annular space (greater than ∼20%, calculated as 100(1 - FT,Rout/FT)). From these considerations, it can be concluded that the maximum expected deviation in the correction factor calculated by eq 37 should not exceed ∼2.5%. Higher deviations may result in poorly designed systems and should be avoided in practice. Experiments 11 and 13-16 provided the highest values for the correction factor and the factor FsT, and high values of L/Rin (recall that L/Rin ) ηL) and L/Rout. From the point of view of the photon absorption process, this is a desirable feature. Furthermore, if more-uniform photon fields are required, then the sets with higher l/L ratios are preferable. Experiments with high Rin values would be also appropriate, provided that L/Rin . 1 (recall that L/Rin ) ηL). The choice of the annular space depth, ∆R ) Rout - Rin, is mainly related to the maximum allowable optical density and to the desired reactor production rate. These considerations may be used as rough guidelines in the reactor design step.

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7597 σλ,g

Napierian spectral extinction coefficient of the inner wall (glass) [m-1]

τλ,c

spectral optical density for all components, eq 28

τλ,j

spectral optical density of component j, eq 40

SE Pabs,λ

spectral volumetric rate of photon absorption averaged over the entire reactor calculated by the LSSE model [mol s-1 m-3]

SE/PP Pabs,λ

dimensionless correction factor defined by eq 37

PSE/PP abs

correction factor averaged over all spectral ranges; ηλ SE/PPλ PSE/PP Pabs,λ Sλk/S abs =∑ k

Subscripts

qj

radiative flux vector [mol s-1 m-2]

λ

wavelength, denotes spectral character (dependence on λ)

r

horizontal distance from the reactor axis to the incidence point [m]

k)1

λk

kth spectral wavelength range conditions at r ) Rin (photon incidence at the reactor inner wall)

Rj

cross-sectional-averaged rate of formation of component j [mol s-1 m-3]

o

Rin

reactor inner radius, i.e., annular space radial boundary [m]

Ω

solid angle (sr)

Ω B

Rout

reactor outer radius, i.e., annular space radial boundary [m]

unit vector in the direction of the photon beam propagation

Ro

distance from the source axis to the inner surface of the inner wall; Ro ) Rin - δg [m]

s

path aligned with the direction at which photons propagate [m]

ηλ S lamp output photon rate (source strength) S ) ∑k)1 λk

Superscripts and Accents f

denotes a vector

o

conditions at the reactor inlet

L

conditions at the reactor outlet

rad

calculated by the radial model

Sλ

lamp output spectral photon rate [mol/s]

s

Sλk

lamp output photon rate emitted in the spectral range λk [mol/s]

refers to the LSSE model (also to spherical surface elements)

v

refers to a per unit volume basis

Sl,λ

lamp output spectral photon rate per unit length; Sl,λ ) Sλ/l [mol s-1 m-1]

yj

cross-sectional average of variable y (except for G h λ,o, which is the axial average of Gλ,o)

tr

residence time; tr ) L/Vjz [s]

yj

average of variable y over the entire reactor

Vjz

cross-sectional average of the mass center velocity [m/s]

VR

volume of the reactor annular space [m3]

Literature Cited

xj

conversion of species j

z

vertical distance from the reactor bottom to the incidence point [m]

z′

vertical distance from the reactor bottom to the emission point [m]

(1) Plaćido, J. Modelagem e simulaç aõ do processo de polimerizaç aõ de metacrilato de metila (MMA) por iniciaç aõ fotoquı´mica, Ph.D. Thesis, Universidade de Saõ Paulo, EPUSP, Saõ Paulo, Brazil, 2000. (2) Alfano, O. M.; Romero, R. L.; Cassano, A. E. Radiation field modelling in photoreactorssI. Homogeneous media. Chem. Eng. Sci. 1986, 41 (3), 421. (3) Jacob, S. M.;. Dranoff, J. S. Light intensity profiles in a perfectly mixed photoreactor. AIChE J. 1970, 16 (3), 359. (4) Romero, R. L.; Alfano, O. M.; Marchetti, J. L.; Cassano, A. E. Modelling and parameter sensitivity of an annular photoreactor with complex kinetics. Chem. Eng. Sci. 1983, 38 (9), 1593. (5) Imoberdorf, G. E.; Cassano, A. E.; Irazoqui, H. A.; Alfano, O. M. Simulation of a mului-annular photocatalytic reactor for degradation of perchloroethylene in air: Parametric analysis of radiative energy efficiencies. Chem. Eng. Sci. 2007, 64, 1138. (6) Imoberdorf, G. E.; Irazoqui, H. A.; Alfano, O. M.; Cassano, A. E. Scaling-up from first principles of a photocatalytic reactor for air pollution remediation. Chem. Eng. Sci. 2007, 62, 793. (7) Puma, G. L.; Khor, J. N.; Brucato, A. Modeling of an annular photocatalytic reactor for water purification: oxidation of pesticides. EnViron. Sci. Technol. 2004, 38, 3737. (8) Spadoni, G.; Stramigioli, C.; Santarelli, F. Rigorous and simplified approach to the modelling of continuous photoreactors. Chem. Eng. Sci. 1980, 35, 925. (9) Hill, F. B.; Reis, N; Shendalman, L. H. Nonuniform initiation of photoreactions. III. Reactant diffusion in single-step reactions. AIChE J. 1968, 14 (5), 798.

Greek Letters δg

inner wall thickness [m]

∇.(∇)

divergence operator of a vector field [m-1] gradient operator of a scalar field [m-1]

φλ,j

spectral quantum yield for the decomposition of species j

ηl

dimensionless lamp length; ηl ) l/Rin

ηlo

dimensionless lamp position; ηlo ) lo/Rin

ηL

dimensionless reactor length; ηL ) L/Rin

ηR

dimensionless reactor outer radius; ηR ) Rout/Rin

ηz

dimensionless axial position; ηz ) z/Rin

κλ,j

Napierian spectral molar absorption coefficient of component j [m2/mol]

F

distance from the point of emission to the point of incidence [m]

σλ,c

Napierian spectral extinction coefficient for all components, eq 3 [m-1]

ReceiVed for reView January 12, 2007 ReVised manuscript receiVed September 4, 2007 Accepted August 31, 2007 IE070078P

2D-LSPP: An Improved Line Source with Parallel Plane Emission

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