Ind. Eng. Chem. Res. 1993,32, 1328-1341
1328
The Multitubular Photoreactor. 1. Radiation Field for Constant Absorption Reactors Ana R. Tymoschuk,+Orlando M. Alfano,* and Albert0 E. Cassano*J INTEC,i Giiemes 3450, 3000 Santa Fe, Argentina
The existing radiation field in a single-lamp, multitubular photoreactor has been modeled from first principles. The device consists of a tubular lamp surrounded by cylindrical tubes of circular cross section, enclosed in a cylindrical reflector. The results show the existence of severe azimuthal asymmetries that cannot be avoided with the inclusion of the reflector. Consideration of the reflected contributions led to a rather complex radiation model. However, using the concept of incident radiation efficiency, the complete model was used to show that, in a multitubular photoreactor designed properly, the reflected contributions should never represent more than 25% of the total; consequently a slightly overdesigned result can be obtained using a much simpler design procedure. It is finally concluded that most probably the use of three-dimensional equations for the thermal energy and mass balances will be unavoidable.
I. Introduction Photochemical reactions have been conducted generally under very mild conditions of temperature and pressure: typically, temperatures between 290 and 400 K and atmospheric pressure. The low operating temperature is one of the strengths of a photoprocess much in the same way that the restriction to low operating pressures may be a weakness. The requirement of a transparent wall made of glass or quartz to permit the provision of radiation inside the reaction space is the origin of the above-mentioned limitation. In some cases, for example in the photochlorination of methyl chloride, several patents describe processes conducted under moderate pressures (Massini, 1984;Holbrook and Morris, 1986). Also, Cabrera et al. (1991) have published bench-scaleresults in a flow reactor, in the liquid phase, and under pressure. There is also one paper that reports a sulfochlorination performed under pressure in a 61-m spiral coil reactor, with an external pressure counteracting the internal pressure of the reaction tube (Boynton et al., 1959). For higher production rates, simpler, continuous tubular photoreactors apt to work under moderate pressures can be used in industrial processes resorting to a multitubesingle-lampphotoreactor. With this purpose, several tubes having a diameter reduced to the point of withstanding the desired pressure, made of glass or quartz, could be arranged forming a pseudoannular reacting space surrounding a tubular lamp. This arrangement has also more degrees of freedom for satisfying heating and/or cooling requirements. Additionally it may include a cylindrical reflector of circular cross section surrounding the tubes (Figure 1). Besides the possibility of working under moderate pressures, multitubular photoreactors offer other advantages. Among them, we can list (i)ample range of operation flexibilities regarding production rates, (ii) simpler routine maintenance with no interruption of the main operation,
* To whom correspondence should be addressed. + Research
Assistant (UNL).
* CONICET Research Staff Member and Professor at UNL.
8 Instituto de DesarrolloTecnoldgico para la Industria Quhica. Consejo Nacional de Investigaciones Cientlficas y TBcnicas (CONICET) and Universidad Nacional del Litoral (UNL).
0888-588519312632-1328$04.00/0
Figure 1. Multitubular photoreactor.
(iii) a lamp system separated totally from the reaction tubes, providing convenient assembly operation, and (iv) under certain conditions, simultaneous, multipurpose operation. Photoreactor design methods have been described recently in several monographic works published by Cassano et al. (19861, De Bernardez et al. (19861, and Cassano and Alfano (1989). No photoreactor modeling for a multitubular photoreactor have been proposed in the open literature yet. In addition to the classical momentum, thermal, and mass balances the design of a photoreactor requires the knowledge of a photon balance accounting for radiative transfer. For a new photoreactor arrangement one must model the radiation field first. This information gives insight to the characteristics of the other balances that must be used when the chemical reaction is incorporated. Thus, if the radiation field presents, for instance, severe angular asymmetries, very likely a two-dimensional mass balance will not suffice. By inspection of Figure 1, this seems to be the case of the multitubular arrangement. This first paper presents a detailed parametric study of the radiation field inside the tubes. A second paper will deal with the presence of the chemical reaction and the corresponding experimental verification. 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 1329
11. Radiation Field In theory, the rate of a photochemical reaction can be described precisely if one knows the reaction mechanism, the specific reaction rate constants, and the primary quantum yield (number of effectively produced photonactivated species, per photon absorbed) (Clarihet al., 1988; Alfano and Cassano, 1988a,b). Usually this information is not available fully. In some simple cases a similar result can be obtained by making use of the overall quantum yield (number of moles of the desired product formed, per einstein absorbed). This is a more phenomenological approach and generally can be used safely under restricted operating conditions (De Bernardez and Cassano, 1985; Vicente et al., 1990; Esplugas et al., 1990). In both cases the rate of initiation (when the primary quantum yield is used) or the rate of reaction (for the case of using the overall quantum yield) is expressed in terms of the quantum yield times the local volumetric rate of energy absorption (LVREA). In other cases the dependence with the absorption of radiation is taken into account through an even more empirical approach, following a power law approximation kinetics which, besides concentrations of reactants (and sometimes products), includes the LVREA. In any event, no matter what type of kinetic approach is adopted, the quantitative treatment requires the value of the LVREA. The LVREA must be written in terms of the spectral incident radiation which, in its turn, can be obtained from the point values of the spectral specific intensity. For polychromatic radiation an integration over the wavelength range of interest is always required. Thus In eq 1 0 is the solid angle of the radiation pencil about the direction of propagation, at the point of incidence I. The LVREA so obtained is then used to write the reaction rate expressions that must be incorporated into the mass balances. It is clear that a previous knowledge of the incident radiation field (G) inside the reactor is important, particularly to study the effect of the different design variables on the radiation distribution inside the reactor. This distribution can be investigated progressing from empty (diactinic) reactors to reactors where the absorption coefficient is constant and ending up with reactors having a variable absorption coefficient as a consequence of changes in concentration produced by the chemical reaction. In the last problem there is a link between the radiation and mass balances that transform the mathematical description of the problem into one of an integro-differential nature (coupling of the radiation absorption with the reaction extent). Diactinic reactors represent the limiting situation for reactors with very low radiation absorption, and constant attenuation reactors are, in general, a good representation of many important photosensitized reacting systems. In the next sections we will present radiation field results for diactinic and constant absorption coefficient reactors, beginning with a single tube and extendingthe results to the multitubular arrangement. The system under consideration includes a reflector surrounding the reaction tubes. In all cases the considered reacting medium is homogeneous. Thus an additional, well-justified assumption is made: there is no scattering of radiation in the whole region under analysis. A chemically reacting system is analyzed in a follow-up paper (part 2). In both the theoretical model and the experiments a well-known actinometric reaction will be employed; in this case, the reaction also occurs in a homogeneous liquid phase (a nonscattering medium).
Reactor
Figure 2. Single-tube system.
111. Single Diactinic Tube Let us consider a single tube of radius m,separated a distance a from a lamp of radius r L , placed inside a cylindrical reflector of radius rn (Figure 2). A point of incidence I(rI,&,zI) located inside the reactor will be considered. Radiation can arrive at point I directly from the lamp (ray D for example) or indirectly from the reflector (rays 1,1and 1,2 for example). To study the radiation field the following assumptions are made: (i) The lamp is a perfect cylinder which emits in all directions from its radiating volume (voluminalemission). If the lamp is of the fluorescent type, the change to superficial emission can be done in a straightforward manner (Cassano and Alfano, 1989). (ii) The reactor is a perfect cylinder and no refraction effects are considered a t its walls. The inclusion of reflection and absorption by the wall is a simple procedure, but refraction effects are much more complicated to account for. No attempts have been made to account for the mismatch of refractive indexes along the ray trajectory; it would involve a complex and cumbersome task. This problem was addressed once by Gebhard (1978) and also studied by Jardz-M. and G a l b (1985). The sound approach proposed by Gebhard cannot be applied to the multitubular photoreactor within reasonable limits in the computerprocessing time; besides, from the limited results included in Gebhard's work, it seems that for short distances between the lamp and the reactor (as in our case) corrections for refraction should not produce important changes in the prediction of the actual radiation field. Given the difficultiesalready encountered in solving the multitubular reactor without including refraction effects, it was considered a workable approximation to think of the reactor boundaries as mathematical surfaces having, at the most, a diminution of transmission produced by absorption (very small if the reactor walls are made of quartz) and backward reflection calculated using Fresnel's formula. (iii) The reflector is a perfect cylinder with specular reflection. (iv)The top and bottom parts of the cylindrical reflector do not reflect radiation. (v) The space surrounding the lamp (from the lamp wall to the reactor wall) is transparent. (vi) Only the first reflection a t the reflector wall produces a significant contribution to the value of the radiation
1330 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993
as
The reactor wall transmission has been taken into account using the transmission coefficient of the quartz tube (TQ), and K~ is a parameter that depends on the characteristics of the radiation source: K,
= E J4r2rL'LL
(3)
The integration limits are obtained following the same reasoning as the one originallyused by Irazoqui et al. (1973) for an annular type of reactor (see Appendix A). After integration for the p and 8 coordinates the resulting equation is GD,u=
+----a
Figure 3. Direct radiation.
field at point I; Le., successive reflections after the first are neglected. This assumption has been shown to be a good approximation in different systems (Cerdl et al., 1977), and it will be even more valid when the optical thickness of the medium is large and when the maximum possible number of reactor tubes are included in the system. The additional effort that would imply the inclusion of more reflections after the first does not seem to be justified. One should consider that in most of the reacting systems of practical interest, when they are well designed, the following conditions will prevail: (i) a radiation ray will travel through a radiation path having from intermediate to high optical density and consequently will be mostly absorbed and (ii) several (as much as possible) reactors will be placed around the lamp to increase the process production and, consequently, the majority of the outgoing radiation from the lamp will never reach the reflector. However, one must be aware that in the case of diactinic reactors, that as a limiting case are analyzed below, the assumption involvesonly a reasonable approximation. To model the radiation field a t any point inside the reactor, all contributions from every elementary volume of emission of the lamp must be taken into account. To do it one can write a radiation balance inside the lamp. Firstly one must obtain an equation that expresses the emission from an elementary volume of emission of the lamp volume to any point in the surrounding space; then one must integrate, for every point inside the reactor, all the radiative contributions from the whole lamp volume. For tubular cylindrical lamps a rigorous three-dimensional emission model, the extense source with voluminal emission model (ESVE) was developed by Irazoqui et al. (1973). The model has proved to predict precisely the radiation field inside a variety of reactor geometries (De Bernardez and Cassano, 1985;Clarid et al., 1988;Alfano and Cassano, 1988a,b; Vicente et al., 1990; Esplugas et al., 1990; etc.). Although the model is conceptually simple, complications arise when the required mathematical expressions for the integration limits must be obtained, particularly if reflecting surfaces are present (see, for example, Cerdd et al. (1977)). For the system described in Figure 1, as stated before, two different types of contributions must be considered: (a) direct radiation and (b) reflected radiation. Direct Radiation. Using a spherical coordinate system located a t the point of incidence I(r~,&,q)inside the reactor (Figure 31, the spectral incident radiation can be expressed
~TR,#,J~FD(~) AeD(4) d4
(4)
with
Reflected Radiation. A0 explained before, only the first reflection will be considered. For reflected radiation, Cerdl et al. (1973, 1977) have shown that GRf,u
= TR,uKurRf,uJ~~~sin
e dp de d4
(5)
Here an additional coefficient, the reflection coefficient, has been included. The indirect incident radiation may be divided into two separate contributions (Figure 2): (1) radiation reflected by regions of the reflector located near the reactor (for example, point Rf1) and (2) radiation reflected by regions of the reflector located in the opposite part of the cylinder (for example, point Rfz). Let us call these regions region 1 and region 2, respectively. Thus, one can write GRf,u
GRf,l,u
+ GRf,Z,u
(6)
The limits of integration for reflected radiation are somehow more difficult to obtain (Figure 4). The methodology is similar to the one described in details by Cerdd et al. (1977)for elliptical reflectors (see Appendix B). After integration for the variables p and 8 the indirect radiation is given by
where
The numerical solution of eqs 4 and 7 was made using the Simpson quadrature method. The computer program automatically and progressively increases the number of integration intervals (always an even number of them); this is done until the result from the last integration differs from the previous one by just a small predefined relative error. This type of integration algorithm is used in all the
Ind. Eng. Chem. Res., Vol. 32, No. 7,1993 1331 Table XI. Diactinic Reactors: Relative Contributions of Incident Radiation. reactor radius distance a % % % % (cm) (cm) Gw.1 G F S ~ Gru GD 28.24 26.05 54.29 45.71 0.5 5.0 0.5 15.0 32.10 24.68 56.79 43.21 0.5 25.0 36.29 23.32 59.62 40.38 29.36 25.63 54.99 45.01 2.5 10.0 31.31 24.95 56.26 43.74 2.5 15.0 35.43 23.59 59.03 40.97 2.5 25.0 Reflector radius = 30 cm. Values correspond to the following spatial coordinates: Q = m;f l ~= T ; ZI = LR/2 = 75 cm.
-1
a1 -
/Ref lector
Reactor
Lamp
(b)
(C)
Figure 4. Reflected radiation. Table I. Photoreactor Parameters lamp0 type: mercury, low pressure arc length lamp diameter cooling jacket diameter input power output nominal power at 254 nm total output nominal power emission characteristic reactor reactor length reactor diameter reactor wall reflector reflector length reflector diameter reflector type average reflection coefficient
150 cm 2.54 cm 5 cm 150 W 45 w 50.1 W voluminal 150 cm 1-5 cm quartz6
150 cm 15-60 cm aluminumc 0.65
0 Data correspond to a commercial germicidal lamp. b Suprasil quality. ALCOA lighting sheet, Alzac treatment.
quadrature procedures employed in the paper to solve the different proposed models (sections V and VII).
IV. Results for the Single Diactinic Tube The description of the system under study is summarized in Table I. In order to avoid unnecessary complications, an almost monochromatic lamp has been chosen; it corresponds to a commercial low mercury pressure, germicidal type lamp having 150 W of input power. The extension to polychromatic light does not involve additional conceptual difficulties, and the general procedure has been described elsewhere (ClariS. et al., 1988). The reactor diameter was allowed to change from 1 to 5 cm; these limits have been chosen considering sizes corresponding to reacting systems having high and low radiation absorption coefficients. In order to analyze the effect of changing the distance from the lamp to the reactor tube, the reflector radius was allowed to change from 7.5 to 30 cm. The reflector was assumed to be made of aluminum, specularly finished, with Alzac treatment (ALCOA,1964).
The results obtained with a single diactinic reactor can be used to analyze two different aspects: (i) the relative contributions of the direct radiation and the maximum possible reflected radiation and (ii) the minimum expected spatial asymmetries. The first aspect is illustrated in Table 11. Direct radiation accounts for approximately 40-46 % of the total incident radiation depending on the distance between the reactor and the lamp. Reflected radiation provides the rest to complete 100%. Indirect radiation from the region of the reflector closer to the reactor (region 1)always produces a more significant contribution than the one coming from the opposite side; these differences become larger when the distance from the reactor to the lamp is increased. The reactor radius does not have a significant effect in these percentage contributions. This rather moderate contribution of the reflected radiation as compared with that produced by the well-known reflector of elliptical cross section, for example, may be explained by considering the shape of the adopted reflector. In fact, a cylindrical reflector of circular cross section concentrates radiation in its own center line. Moreover, it must be noticed that these values for the reflected radiation constitute an upper limit contribution. If one considers a single non-diactinic tube, this contribution will decrease as a consequence of the shadow effect that the tube produces on the reflector (decreasing the reflected radiation from region 1). In addition, if one considers the existence of several (N),non-diactinic tubes, the reflected radiation will be further decreased by the shadow effect produced by the remaining N - 1 tubes on the reflector capabilities with regard to the one tube under analysis. Important spatial asymmetries are observed even in diactinic reactors where the existence of significant reflected radiation tends to make more uniform the irradiation from all directions. Figure 5 depicts the results for the 2.5-cm reactor radius. It can be noted that, as expected, the reactor has one plane of symmetry defined by the diameter in the 0-r position and the z axis (Figure 5b) and a second one defined by the plane perpendicular to the reactor axis intersecting the reactor at the middle high position (Figure 5c). On the other hand, angular asymmetries and longitudinal nonuniformities are important. The incident radiation at 61 = r (point of the reactor closer to the lamp) is 23% higher than that at 61 = 0 (computed at rI = 2.5 cm and ZI = 75 cm, and with respect to the larger value). Similarly, for I-I = 2.5 cm and @I = r , the difference between the incident radiation at ZI = 0 cm (or 150 cm) and ZI = 75 cm is about 40% (computed with respect to the larger value). Finally, it can be observed that the incident radiation takes on the largest values in the second and third quadrant (from 61 = r / 2 to 61 = 3 ~ 1 2always, ) while the lowest values occur in the first and fourth quadrant. This is just a direct result of the proximity of the first two regions to the lamp. One must note that the axial variations predicted by the model are not originated by "emission end effects" in
1332 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 12 v)
N
E
r R f = 30
cm
\
.-E e
L
VI
.-C e
m
H/2
10
2
-
x
& 09 &
2 50
0
125
2 50
125
those observed for a single tube. When one includes N 1additional tubes in the system, there will exist interaction among the tubes; these interactions may be analyzed by studying the changes produced in the limiting angles of integration on a plane defined by the (r,B)coordinates or, which is the same, the limiting angles of integration for 6(refer to section VII). On the other hand, axialvariations are associated with changes along the (z,j3) plane. For this particular effect, only the limiting angles of integration for 6 are affected by the above-mentioned geometrical relationships (lamp length versus reactor length); Le., the inclusion of more tubes does not affect the limits of integration for the 6 angle. V. Single Tube with Constant Absorption
-
“E
a-
I
/
1.00
0
\
I
t
I
0
50
1
zr (cm 1
100
J
The previous analysis shows the radiation field established by the lamp inside the reaction space of an empty reactor. As may have been expected, some azimuthal asymmetries were found but they are not larger than 25 7%. On the other hand, axial variations are more important. In approaching amore common situation, the next problem to consider is the study of the radiation field when radiation absorption occurs inside the reactor; particularly, its effects on the observed asymmetries. A system having a constant absorption coefficient will be analyzed, which comprises a wide range of photosensitized reactions. The general equation representing the incident radiation generated inside the reactor volume by the three-dimensional emission of the lamp can be obtained by combining the emission model (ESVE model) with a radiation balance inside the reactor. Generalizing previous developments presented by De Bernardez et al. (1987) and Cassano and Alfano (1989), one can write
I50
(C)
Figure 5. Incident radiation energies in a diactinic reactor. (a) As a function of the reactor radius; angle PI is the parameter. (b)As a function of the angular position; radial position rI, in cm, is the parameter. (c) As a function of the longitudinal position. Values are for a = 20 cm, rn = 30 cm, rI = 2.5 cm, and PI = A.
the source. The ESVE model assumes that emission from the tubular lamp is constant (uniform and steady) along its axial direction. A detailed way of accounting for end effects associated with the lamp emissionhas been carefully analyzed in a previous paper by De Bernardez and Cassano (1985). This work, obviously, does not take into account some anomalous performances (outside nominal specifications) of commercial lamps, as found in one part of the experiments reported in Gebhard’s work (1978). Hence, the differences that are shown in Figure 5c for the values of G are originated in the geometric characteristic of the whole system (particularly resulting from the ratio of the lamp length to the reactor length). For example, an infinitely long lamp-with respect to the reactor lengthwould produce an almost uniform radiation field in the axial direction; conversely, a very short lamp would produce verysignificant changes in the radiation field along the z direction. The phenomenon was first studied by Jacob and Dranoff (1969) and later on analyzed by CerdB. et al. (1973) for the case of a cylindrical reactor inside a cylindrical reflector of elliptical cross section, and by Irazoqui et al. (1973) for an annular type of reactor. For a multitubular reactor one can anticipate that changes along the axial coordinate will be equivalent to
where now, in the exponential attenuation, the integration along the radial spherical coordinate must be performed inside the reactor. This equation, when properly applied to each particular geometry, provides the local value of the incident radiation at every point inside the reactor. When eq 8 is used to obtain the contributions of the direct radiation, the limits of integration previously described for a diactinic tube (section 111) can be used without changes. Hence, for direct radiation the following result is obtained:
In eq 9 , 6 and ~ are given in Appendix A. F D ( ~is)given by eq 4.a. Indirect radiation can be obtained in a similar manner, and again the limits of integration previously derived in section I11 are valid. By proper inspection of the limits for the variable 4, it is possible to discriminate again between the two regions of reflection in the cylindrical reflector. The resulting equation is GRf,v
=
2 ~ R , ~ ~ ~ ~ R f , u ~ b ~ ~ RPRO~ ( ~ dp*)l ) ~ ~ bde, de 4~ (10) p [ ~ p R ’ - ( ~ u
In eq 10, and 4 ~are f given in Appendix B. F&$) is given by eq 7.a. For the particular case of a constant absorption coefficient inside the reactor, the integral in eqs 9 and 10 can
Ind. Eng. Chem. Res., Vol. 32, No. 7,1993 1333 Table 111. Reactors with Constant Absorption Coefficient: Relative Contributions of Incident Radiation. reactorradius (cm) 0.5 0.5 0.5 2.5 2.5 2.5
absorptioncoeff (cm-1) 0.1 1.0 10.0 0.1 1.0 10.0
% Gw.1 27.64 11.56 0.103 18.63 1.148 0.023
% Gw2 26.26 32.11 36.29 29.53 35.24 36.27
%
%
Gw GD 53.90 46.10 43.67 56.33 36.39 63.61 48.16 51.84 36.39 63.61 36.29 63.71 a Reflector radius = 30 cm; distance a = 10 cm. Values correspond to the following spatial coordinates: rr = m; @I = r ; ZI = LR/2 = 75 cm.
the third situation. The ray is absorbed partially from b to IO’,is then reflected at the point Piii, and is absorbed finally as reflected radiation, from IO”to I. In terms of intensities we have
Since
I Figure 6. Attenuation of direct radiation.
we obtain Equation 15 can be expressed in terms of the incident radiation G by integrating over the solid angle of radiation reception. In eq 15 Ap is given by eq 12. If ApR is less than or equal to zero, we are in the second situation. The value of APR is given by Ref lector/
Figure 7. Attenuation of reflectad radiation. (For the drawing, rays have been projected on a plane perpendicular to the reactor
where
axis.)
02 sin2 5 =
be written as
[(x,
+ rI cos &)sin 4E- bP+ rI sin flI)cos$El2
(17)
VI. Results for a Single Tube with Constant Absorption In this equation one must evaluate Ap from the point of entrance of the ray (at the reactor wall) to the point of incidence under consideration. Three different situations can be encountered (Figures6 and 7): (i) radiation arriving at the point of incidence I in a direct way, (ii) radiation arriving at the same point after one reflection and that has not been previously intercepted by the reactor, and (iii) radiation arriving at point I after one reflection, correspondingto rays that had previously transversed the reactor (asdirect rays) and have been reflected in regions that correspond to umbral and penumbral zones of the reflector. Rays corresponding to situation iii must be excluded from the computation of the reflected incident radiation only in those cases in which the rays are absorbed totally in their first passage through the reactor (see for details, and for the case of an elliptical reflector, CerdB. et al. (1977)). The fist situationis illustrated in Figure 6. The general expression for Ap can be applied directly:
-r, cos(4 - pI) + [ r 2 - r: sin74 - 01)~1/2 (12) sin B The second and third situations can be analyzed by inspection of Figure 7, where reflected radiation from region 1 is shown. The more general case corresponds to Ap =
As was done before, the obtained results for a single reactor tube with a constant absorption coefficient can be used to analyze two different aspects: (i) the relative contributions of the direct and the reflected radiation and (ii) the spatial asymmetries. Table I11 shows that the presence of significant absorption produces drastic changes in the available radiation field. An absorption coefficient of 0.1 cm-l is an almost diactinic reactor (to absorb 99% of the incoming intensity, it would require a radiation path of about 46 cm). The other extreme may be represented by a reactor of very high absorption. In this case most of the intensity will be absorbed in a very thin film adjacent to the reactor wall (for example p = 100cm-1 would require a radiation path of just 0.046 cm), and the surface of this reactor would act almost as a blackbody. Very likely most practical reactors will use design values of p ranging from0.25to 10cm-’ depending on the reacting system and reaction conditions. Analyzing the relative values of the three contributions, the behavior of the small-radius reactor is similar to the one with a larger radius. Additionally, in both cases when the absorption is increased beyond a value of p = 0.1 cm-l, the relative contribution of the reflected radiation coming from region 1 of the reflector is significantly decreased and for p = 10 cm-l is almost null. This progressive
1334 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993
Figure 8. Incident radiation energies in an absorbent reactor for a = 20 cm, = 30 cm, and r~ = 2.5 cm. (a) As a function of the radial position for p = 0.1 cm-1; angle @I is the parameter. (b) As a function of the angular position for p = 0.1 cm-l; radial position n,in cm, is the parameter. (c) As a function of the radial position for p = 50 cm-1; angle @I is the parameter. (d) As a function of the angular position for p = 50 cm-l; radial position n,in cm, is the parameter.
weakening of the reflected radiation from region 1is due to the shadow effect produced by the radiation-absorbing reactor on those parts of the reflector that could had been useful for conveying reflected radiation on the same reacting tube. This is not the case of the reflected radiation from region 2; in this case, the reflection capabilities of the reflector are not altered by the presence of a single absorption tube. The effects of the species radiation absorption on the radiation field inside the reactor tube can be made more evident when the spatial variations of the incident radiation are analyzed systematically. It is convenient to study the cases of low and high radiation absorption separately. Results are always shown for a fixed value of the axial position (21 = (1/2)LR). Firstly, the case of low absorption ( p = 0.1 cm-l) is illustrated in a plot of G vs n,using some selected values of 01as a parameter (Figure 8a). Even in this case of low absorption, the values of G for PI = A and PI = 0 are very different; in fact at the reactor wall for 61 = 0 the value is only approximately 60% of that for P I = A. Under low absorption conditions the plot of G vs 01(using rI as a parameter) shows an unexpected result: the maximum value of G is not obtained at PI = A, but at two symmetric 2 8b). This positions located near 01= ?r/2and 3 ~ 1 (Figure phenomenon is caused by the reflected radiation. The reflected radiation coming from region 1 originates a radiation field of some importance on the first and fourth quadrants and a much weaker one on the second and third quadrant. Conversely, the reflected radiation proceeding from region 2 is stronger in the second and third quadrants and almost null in the first and fourth. When the reactor has low absorption, there will be a significant overlapping of both effects in some regions of the reactor surface. Under these special conditions these two effects added together generate the observed maxima at P = a12 and P = 3 ~ 1 2 . This type of behavior, characteristic of low-absorption reactors, was not observed in the diactinic tube. Results for reactors of very high absorption ( p = 50 cm-') are shown in Figure 8c,d. It is not surprising to find that a large portion of the reactor becomes useless: only the outer part of the tube is active as far as the photochemical reaction is concerned and, what is also important, those regions located in the vicinity of the
reactor boundaries defined by 0 I 01I ?r/2 and 3 ~ 1 I2 61 I 27r receive less than 50% of the incident radiation arriving at the opposite side. When the absorption is high, the maximum incident radiation is observed at PI = A. This value is more than 20% smaller than the one obtained with p = 0.1 cm-l at the same position, as can be deduced from the results shown in Figure 8b,d. In this case, peaks are observed 2 6 = 3 ~ /as 2 was explained also in regions close to 6 = ~ / and before; however, now we are analyzing a case where p is high. Under this circumstance, the direct radiation and that reflected by region 2 prevail and the maximum on the curve appears for 0 = A. These results indicate that even if one works with a single tube, the asymmetries are severe. Moreover, the inclusion of a reflecting surface surrounding the reactor tube does not make a significant contribution to improve the system azimuthal symmetry. Parametric Analysis of the Angular Asymmetries. Angular asymmetries are unavoidable in a reacting system like the multitube reactor. However,geometricparameters such as the reactor radius (plt), the reflector radius (m), and the lamp-reactor distance (a), selected for the most appropriate design, may contribute to reduce, if not eliminate, its effects. In order to study these design parameter influences on the radiation angular distribution, the following dimensionless variables are defined:
The value of A gives an indication of the reactor azimuthal asymmetry. We call A the dimensionless asymmetry ratio. Note that both values are defined at the reactor surface. Using the fixed value of the lamp radius (a)as a characteristic length, the remaining parameters are
The value of rr, is adopted from the standard size of a commercial lamp and is equal to 1.27 cm (Table I). When the dimensionless asymmetry ratio is plotted against the distance from the reactor to the lamp (yk), a very distinct behavior is observed for reactors of small radius (YR = 0.4) and for a larger reactor tube radius (YR = 2). As shown in Figure 9a for p = 0.1 cm-1 (low absorption), the value of A first decreases, passes through aminimum, and then increases. This behavior is repeated for different values of yw. The asymmetry reaches its lower value (less than 0.1) for the largest value of the reflector radius. With different numerical results, this behavior was found regardless of the value of the absorption coefficient. In fact, for very high absorption ( p = 60 cm-I), curves not plotted here show an analogous shape passing through a minimum value of A. However, for this case of very high absorption, the dimensionless asymmetries in the radiation field are stressed significantly. These results, the existence of a minimum and the striking difference between the values of these minima, have different causes. The existence of a minimum for a reactor of small diameter is caused by the changes in the relative contributions of direct and reflected rays when the distance from the lamp to the reactor (and consequently the reflector radius) is changed. On the other hand, the persistence of very large
Ind. Eng. Chem. Res., Vol. 32, No. 7,1993 1335 I
0.4
t m
0
0
0
10
20
30
40 01 0
I
1
I
I
10
20
30
40
50
40
50
p (cm-') (a1
m
E
.01
In
.-
0
m
9 05 X
r\' 0.8-
P)
YR =
V
0 0
IO
20
30
p ( crn-') (b)
Figure 10. (a) Total value of VREA vs absorption coefficient. (b) Average value of VREAvs absorption coefficient. Values are for = 20cmanda = 10cm.
values of A under high absorption conditions indicates that no compensation effects can overcome the complete absence of direct radiation at j3 = 0 when the attenuation of the incident radiation is so strong. When the reactor radius is larger, the important differences between the dimensionless asymmetry values for low- and highabsorption systems are maintained. However, as can be observed in Figure 9b, the minimum in the curves of A vs ya for practical values of y ~does f not exist. Once more this behavior is independent of the absorption characteristics of the system and the dimensions of the reflector. When the reactor radius is sufficiently large, the umbral and penumbral zones at the reflector surface in region 1 of reflection are so significant that under no circumstances can the reflected radiation provide a significant amount of incident radiation. Consequently regions of the reactor in the neighborhood of j3 = 0 always exhibit poor irradiation. The observed monotonous decreasing in the value of A is mainly the result of the decrease in the direct radiation contribution as a result of the increasing distance from the lamp. It should be noted that the minimum value of A never falls below 0.2 for practical values of YRf. An analogous performance is obtained for high values of p. For instance, for p = 50 cm-1, results not plotted here showed a continuous decrease in the value of A from 0.8 to 0.4 for practical values of yw,with no indications of a significant change in the slope; i.e., for the values of the
geometric parameters used in this work, the minimum cannot be observed. The influence of the value of the absorption coefficient ( p ) on the dimensionless asymmetry ratio is important from p = 0 top = 10cm-l; the asymmetry ratio is not sensitive to changes in the value of p beyond 10 cm-l (Figure 9c). Average Volumetric Rate of Energy Absorption. As indicated in eq 1,the important variable in a photochemical reactor is the LVREA. It seems appropriate to investigate the way in which the averaged value of the LVREA, computed over the reactor volume, changes with the absorption characteristics of the system and the reactor radius. Both effects can be analyzed from the total value of the VREA and the average VREA. This means that we will compute eaT= h 2 r S o L R ~ ~dre adzr dj3
(22)
and
Figure loa shows values calculated according to eq 22. The reactor of larger radius always renders the best values of e%. The total value of the VREA becomes almost insensitive to changes in p beyond values of p = 5 cm-l; Le., above this value of the absorption coefficient, even the small-diameter reactor becomes very inefficient. For practical purposes the results obtained with eq 23 may be more useful (Figure lob); they indicate that the energy is used more efficiently in the smalT-radius reactor. In
1336 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993
practical terms, at constant linear velocity (or constant average residence time), and in the majority of the circumstances, the small-radius reactor will have higher reactant conversion. This better performance is obtained at the price of lower flow rates. For a given requirement in the production rate and in the product concentration, the answer will be obtained after posing properly a complex optimization problem. This is more so because the number of tubes in a multitubular reactor is not only a function ) also of the distance from the of the reactor radius ( p ~but lamp to the tubes (a). This additional variable affects not only the capacity of the device to place a different number of tubes for each tube diameter, but also the amount of incident radiation that reaches the tube (Gdecreases when a is increased).
\ Yt
Reactor 2
- 1
\
\
I
r
rt YP
-
h
L
X
P
-
/
Figure 11. Geometric descriptionof the analysisrequiredfor defining the limiting angles for reflected radiation coming from region 1 in a multitubular reactor.
VII. Multitubular Reactor with a Constant Absorption Coefficient As was described for the reflected radiation in a single diactinic tube, each reactor tube receives indirect radiation from two different regions: one located in the reflector near the tube under consideration (region 1) and the other located in the opposite part of the reflecting surface (region 2). When there is a single tube, under all conditions of absorption we will have the maximum possible contribution of reflected radiation. When we add more reaction tubes, the contributions of the direct radiation remain unchanged while those of reflected radiation, under most practical conditions, will decrease. Exceptions to this situation will be described below. In what follows we will derive a methodology to study systematically the effect produced by the inclusion of an orderly increasing number of reactor tubes on the reflected and total incident radiation. The study will be conducted using the idea of incident efficiency proposed first by Cerdl et al. (1977). Number of Tubes of the Multitubular Reactor. Let us assume that the reactor tubes are distributed in the circumference of radius a in an uniform fashion separated by a distance defined by an angle w1. This separation will be dictated, at least, by two practical considerations: (i) reactor assembly and (ii) heating or cooling requirements. If
then
where ELis the output photonic power of the lamp, q is the radiation density flux vector, CI is a unit vector aligned with the radiation beam but oriented in the opposite direction, and il is the solid angle about q. The value of 71 is a direct measurement of the ability of any reactor system to collect the radiation produced by the lamp regardless of the absorption characteristics and the kinetics of the reacting system. Hence, it can be used to analyze the efficiency of the multitubular reactor as a function of the number of tubes and the tube diameter. Direct Radiation Density Flux. According to eqs 30 and 31, the direct radiation density flux is given by
where
cos 8, = sin 8 cos(4 - PI) (33) and 8, is the angle between -q and the inwardly directed normal vector (n) to the surface of the reactor. The limits of integration are given in Appendix A. After integration for p and 8: qD.u
-~ T R , , K , J ~ D (cod4 ~ ) - 41)[cos &),I
and
- COB $,a]
d4 (34)
(26)
Reflected Radiation Density Flux. Applying eqs 30 and 31, the reflected radiation density flux is
where h is the angle measured at the center of the multitubular reactor that defines the separation between two adjacent reactor tubes (Figure 11). Incident Efficiency. The overall efficiency of a photoreactor may be described by the product of four separate yields:
The limits of integration are given in Appendix B. Again, integrating for p and 8:
N R = 2(*/wl)
1 = TLVIVA*
(27)
where VL is the lamp photon yield, ?')A is the absorption efficiency of the reaction volume, 0 is the overall quantum yield, and 91 is the incident efficiency defined by
qRf,v = ~ T R , , K , ~ R ~ , J ~ Fcos($ W ( ~ )- PI) [cos e ~ f , l -
cos e , , ] d 4 (36) Equation 36,with the limits stated above, applies for a single tube. When more tubes are added, one must investigate the effects produced by the additional tubes on the ability of the reflecting surface to irradiate indirectly
Ind. Eng. Chem. Res., Vol. 32, No. 7,1993 1337 the reactor tube under consideration. In practice, this interference in the reflected rays affects the limits for $ only. The analysis is different for each of the two different regions of reflection that were defined in section 111. In order to reduce the complexityof the analysis,let us assume that the system is well designed and, henceforth, all the arriving radiation a t each reactor tube is absorbed ( 7 =~ 1). This assumption permits a simpler use of the concept of incident efficiency. For the reflector region 1,Figure 11shows the way by which reactor 2 affects the ability of a portion of the reflector to irradiate the point of incidence I at reactor 1. From the details given in the figure, it is clear that one needs to identify, for each point P(xp,yp) on the reflector, a ray like the one indicated by ita intersections E1 and E2 with the lamp. Any other ray above this one will cross reactor 2 and will not be able to be reflected and, afterwards, impinge on some part of reactor 1. In the figure, STindicates the distance from point P to the point where the ray is tangent to the reactor. One can also identify points such as 1and 2, with distances SI and S2, respectively, that corresponds to a ray that intersects reactor 2. The tangent ray satisfies
AS = s, - s, = 0
Lomp
Reactor 1
(a)
(37)
From the figures one gets immediately
s2+ 2ss0COS($E - a0)+ st =; ' ?
(38)
so=
Lamp
[(x,
+ h - a cos wJ2 + Cyp + k - a sin w1)211/2 a sin w1 - (yp + k)] a. = - sin-' A
Reoctor 1
(38.a) (38.b) (b) Figure 12. Geometric descriptionof the analyais required for defining
where xp, yp, k,h, and $E are obtained in Appendix B. The solution of eq 38 will give, in general, two roots. The case in which SI = S2 = STgives AS = 0. Then, the solution of eq 39 equated to zero, which is a function of $ only, provides the position of the ray corresponding to the straight line defined by E1 and E2. This value of $ = h is the new limit for $WJ.Any value of $ between $T and the old value of $ R ~ J must be excluded from the contributions of reflected radiation to reactor 1. This analysis, made with reactor 2 located above reactor 1in Figure 11, can be repeated in exactly the same manner for one reactor located below reactor 1. This second procedure will provide the required changes for $ ~ f , 2 . The result is a set of equations equal to eqs 38 and 39. For the second region of the reflector, i.e., the one placed in the opposite side, once more we must investigate what portions of the reflector are not useful any longer because the possibilities of reflected contributions are precluded by including one or more reactor tubes that intercept the rays. Figure 12a illustrates the problem for a case where four reactors are placed in the system. As far as reactor 1 is concerned, reactor 3 is the only one that interferes with the reflected radiation from region 2 of the reflector. For a point of incidence such as I(r&) the original limiting angles are $3 and $4. Due to the presence of reactor 3, (let us say j = 3), the region of the reflector defined between the angles $j,3 and $j,4 cannot contribute with indirect radiation to the point I. The problem to solve has two steps: (i) to find, for the reactor tube under analysis, how many reactors are placed in the space that interferes with the radiation that should be reflected in region 2 and where
the limiting angles for reflected radiation coming from region 2 in a multitubular reactor.
are they located, and (ii) to find the way to calculate the limiting angles that exclude reflecting portions of the reflector due to the presence of those interfering reactor tubes. Let us assume that we put N R reactors. The position of each of the j reactors (j = 1, 2, ..., NR),expressed in terms of the angular coordinate about the center of the multitubular device and measured from the first reactor, is given by sj = (j - l)ol j = 1,2, ,..,NR (40) Using Figure 12a as an example, one must find the tubes or portions of the tubes that interfere with the possibilities of producing reflected radiation upon reactor 1. Reactor 3 (j = 3) located at r j = A is the interfering tube. The original region of reflection from A to D is now reduced to a fraction from A to B and another from C to D. The contributing reflected rays will now have trajectories that lie between $3 and $j,3 and $j,4 and $4, with j = 3. For the general case, where more than one reactor can mask the reflector, the same procedure can be employed. We must obtain the set of equations required to compute the angles $j,3 and $j,4; they define the region of shadow that the j reactor produces on the reflector in all what concerns the point of incidence I. From Figure 12b, with geometric considerations, it can be obtained that $j,$ = 7F + a + 7j - Si (41) $j,4 = A + a + 7j + Si (42) where ~j and Sj can be calculated from the following
1338 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 0.03
equations: rj = c o d
(42.a)
6 = sin-’(rR/Zj)
(42.b)
Z j = [a2+ a2- 2aa cos-l(tj - a ) ] ’ / 2
(42.c)
The angle 5;. can be computed from eq 40 according to the chosen value for j and the adopted angle of separation between reactors ( ~ 1 ) .Finally, a can be obtained from eq A.5. Note that, according to the location of the j reactor under analysis, the angle T j may be greater than zero (as is the case in Figure 12b), equal to zero (when the point J lies on the line defined by the direction E),or less than zero in the case where the point J is located below the line defined by the previously mentioned direction. In short, for apoint of incidence such as I, the calculating procedure is (i) compute the original limits 43 and 4 4 (with no interferences from the other N - 1 reactor tubes); (ii) using eqs 40, 42.c, 42.a, and 42.b, compute [ j , Zj, rj, and 6 j , respectively, for each of the possible j interfering reactors; (iii) using eqs 41 and 42 compute 4j,3and 4 j , 4 ; (iv) eliminate from the original integration interval 143, 441 calculated in i the interference interval [ 4 j , 3 , 4 j j , 4 ]obtained in iii; and (v) repeat steps ii-iv for every of the j values until all possible j interfering reactors have been considered.
0.02
0 01
I
0
With the theory developed in section VI1 it is now possible to study the performance of the multitubular reactor using the concept of incident efficiency according to eqs 28-31,34, and 36. For eq 36 the new limits for 4 are obtained using eqs 38 and 39 for reflecting region 1 and eqs 40-42 for reflecting region 2. The analysis can be done by changing N R in eq 26 and correspondingly in eq 40 from j = 2 to j = NR,where NR is the number of tubes distributed in the circumference of radius a. The value of NRis limited by the maximum number of tubes permitted by the circumference of radius a. Let us first look at the effects produced in only one tube, when we increase the number of tubes in the system. This inspection will be done for the incident efficiency of the direct radiation and the total incident efficiency.Thus, in eqs 28 and 29, we will use for ~ I , Dthe results provided by eq 34 and for 7 1 , ~the sum of the results provided by eqs 34 and 36. The results will be shown for a multitubular reactor of the characteristics described in Table I. In what f be also follows, yawill be constant and equal to 8, y ~will constant and equal to 16, and two reactor radii will be analyzed: YR = 0.4 and YR = 2. When only a few reactor tubes are added to the system, the reflected radiation contributions from region 1(which, from the assumption of high absorption made in section VII, are always small) remain unchanged. The contribution of the reflected radiation from region 2 changes drastically depending whether the number of added reactors is even or odd. Figure 13 illustrates these results. When only a few reactors are present, the angular separation between them is large; consequently, when the total number of reactors uniformly and symmetrically distributed in the circumference of radius yais odd, there is no effect on the reflector abilities to increase the amount of radiation arriving at reactor 1. On the contrary, with
I
I
40
I
I
60
NR (a)
710’15
I
0.05
0
VIII. Results for t h e Multitubular Reactor with Constant Absorption Coefficient
I
20
0
4
12
NR (b) Figure 13. Incident efficiency computed for only one tube for Y. = 8 and yw = 16. (a) TR = 0.4. (b) YR 2.
only two reactors in the system, most of the hypothetical reflecting region 2 for reactor 1is masked by the second reactor. Keeping in mind that we are analyzing here the resulting effect on a single reactor tube, the condition of a small and odd number of reactors provides the maximum total incident efficiency. When the number of reactors is increased beyond a number which is a function of Y*, y ~ f , and YR, the effect of the added tubes on the ability of reflecting region 2 is to decrease the reflected radiation, regardless of the odd or even characteristics of the total number of tubes. When this number is sufficiently high, the added tubes also affect the ability of reflecting region 1to convey radiation on reactor 1and the total incident efficiency is further decreased. These two effects, when added, produce a variation of VI,T as a function of N R in a more “continuous” fashion. Figure 13 also illustrates that, computing the incident efficiency for only one tube, the direct incident efficiency shows a constant value, independent of the number of tubes included into the system, which corresponds to the minimum expected value for TI,T. Figure 14 permits the analysis of the effect of N R on the whole multitubular reactor system. The direct incident efficiency increases linearly with the increasing number of reactors. The total incident efficiency also increases with increasing NR, but the function has a serrated form. This behavior is the result of the alternate contribution produced by the addition of an odd or an even number of reactors, according to the observed performance for only one tube. In theory, leavingaside some losses a t the top and bottom of the system due to the spherical characteristics of the
Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 1339 1.0 r
1
t
0.0
t
#
0
0.6
1
0'4 0.2
0.0 0
20
60
40
NR (a) 1 .o
0.0 0.6 0.4
0.2 /
0.0 0
I
I
4
I
1
0
I
I
12
NR (b) Figure 14. Incident efficiency computed for the whole multitubular system, for ya = 8 and y~ = 16. (a) YR = 0.4. (b)7~ = 2.
radiation emission, when the whole circumference corresponding to Y~is filled with reactor tubes, 7% approaches 1. This is the incident efficiency corresponding to an annular type of reactor. Under these conditions, the reflector is useless. It should be noted, however, that practical considerations do not allow placing reactors with their wall in close contact. Under normal assembly conditions the separation between reactors will permit a maximum percentage contribution of the reflected radiation in the order of 2025%. These figures may be deduced by considering a reasonable large number of reactors surrounding the lamp (in order to use as much as possible the emitted energy) but allowing enough room between reactors to permit the normal operations of assembling and maintenance. For instance, in Figure 14a NR I62; if 40-50 reactor tubes are installed, the system will render a reflected radiation contribution that may range from 15% to 26 % of the direct radiation arriving to the reactors. Similarly in Figure 14b N R I12; placing from 8 to 10 reactor tubes, the system will produce a reflected radiation contribution ranging from 20% to 29% of the radiation arriving directly. Under no circumstances, with the proper choice of the reactor diameter to ensure totalabsorption of the incoming radiation, would a second row of reactor tubes in a circumference of radius (rat> ?a) be advisable. The necessary separation between tubes of the first row does not provide enough unused space to permit efficient irradiation of the second row, and besides, the increased distance from the lamp (Tat > ya) reduces even further the radiation density flux impinging at the second row.
IX. Conclusions A complete model for the radiation field inside multitubular photoreactors of constant radiation absorption coefficient has been developed. From the model the following can be concluded: 1. In accordance with previous results for other photochemical systems, for example, the annular photoreactor (Irazoqui et al., 1973), the tubular reactor inside an elliptical reflector (Cerdl et al., 19731, the cylindrical reactor irradiated from the bottom (Alfano et al., 1985) and the single-tube, multilamp photoreactor (Alfano et al., 1990), the radiation field inside the reactor is not uniform along the direction of the tubular lamp axis. 2. Unlike all the other systems, there exist no conditions under which azimuthal asymmetries can be avoided. For instance, in the annular photoreactor they are nonexistent; in the cylindrical photoreactor irradiated from the bottom as well as in the single-tube, multilamp photoreactor their magnitude is generally negligible; in the cylindrical reactor with an elliptical reflector, there exist geometrical conditions which can make the angular asymmetries insignificant. In the multitubular photoreactor the angular asymmetries are always important. 3. The inclusion of a reflector surrounding the tubes does not contribute significantlyto produce a more uniform radiation field as far as the angular coordinate is concerned. If one had been considering more than one reflection at the reflector walls, particularly for systems of low radiation absorption (a radiation path of low optical density), a slightly more uniform radiation distribution could have been encountered (the angular distribution would have been smoother). However, for these systems the diactinic approximation would be valid and, as shown in the paper, in these cases angular asymmetries are not very significant; i.e., angular asymmetries are very important in systems of moderate to high radiation absorption. 4. The consideration of the reflected rays originates an intricate and rather complex model, particularly in the formulation of the limits of integration for the computation of the photon balance. Through the use of the concept of incident efficiency and the model, it was possible to show that, in a well-designed multitubular photoreactor, the contribution of the reflected radiation should never represent a value larger than 20-25 % of the direct incident radiation. Thus, computation of the reflected radiation for most practical cases may be avoided. This introduces an important simplification in the design, and the resulting reactor will be, at the most, overdesigned by a small percentage. 5. Very likely, under all circumstances, the design of a multitubular photoreactor will require three-dimensional mass and energy balances. This is the direct consequence of the observed severe angular asymmetries in the multitubular system. 6. A good design of a multitubular photoreactor should require the solution of a complex optimization problem. For a prescribed production rate and product concentration, the optimal design from the radiation viewpoint will be the result of the proper combination of the following geometrical parameters: reactor radius, distance from the reactor to the lamp, and number of reactor tubes. All of them are explicitly or implicitly influenced by the absorption coefficient, which, in its turn, affects explicitly the kinetics of the reaction.
1340 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993
Acknowledgment The authors are grateful to Consejo Nacional de Investigaciones Cientlficas y TBcnicas (CONICET) and Universidad Nacional del Litoral (UNL) for their financial aid. Nomenclature a = distance between lamp axis and reactor axis, cm D = distance defined by eq B.4, cm ea = local volumetricrate of energy absorption, einsteins cm3 S-1
E = lamp output photonic power, einsteins s-1 F = variable defined by eqs 4.a and 7.a, cm G = incident radiation, einsteins cm-' s-l h = variable defined by eq B.8, cm I = intensity, einsteins cm-2 s-l s r l I = point inside reactor k = distance defined by eq B.9, cm L = length, cm m = variable defined by eq B.ll, dimensionless n = unit normal vector, dimensionless N = number of reaction tubes q = radiation density flux, einsteins cm-2 s-1 Q = total incident radiation, einsteins s-l r = radius or radial coordinate, cm S = variable defined by eq 38, cm V = volume, cm3 x = rectangular coordinate, cm y = rectangular coordinate, cm z = cylindrical and rectangular coordinate, cm 2 = variable defined by eq 42.c, cm
j = "j" reactor
L = lamp property P = reflection point at the reflector R = reactor property Rf = reflector property or reflected incident radiation T = total value V = volume X = wavelength v = frequency 0 = relative to the surface of radiation entrance 1 = lower limit of integration for region 1 2 = upper limit of integration for region 1 3 = lower limit of integration for region 2 4 = upper limit of integration for region 2 Superscripts
a = absorption N R = relative to the whole multitubular reactor O = relative to the surface of radiation entrance ' = value projected on the x-y plane * = relative to the attenuation path Special Symbols ( ) = average value
Appendix A Integration Limits for Direct Radiation (Figure 3). For the variable p:
- a cos 9 =F [r; - u2 sin241'/' pD,Cl,Z)
Subscripts
a = relative to the distance between lamp and reactor A = absorption D = direct incident radiation E = property of a ray coming from the source I = incident point at the reactor or a property of an incident ray
sin B
(AS)
+ (rIsin &)21'/2
(A.2)
where
Greek Letters a = molar absorptivity, cm2 mol-'; also angle defined by eq
A.5, rad a0 = angle defined by eq 38.b, rad @ = cylindrical angular coordinate, rad y = radial coordinate, dimensionless r = reflection coefficient, dimensionless 6 = angle defined by eq 42.b, rad A = angular asymmetry, defined by eq 18, dimensionless; also angle interval or distance between two points c = unit vector aligned with the radiation beam, dimensionless { = angle defined by eq 40, rad 7 = efficiency, dimensionless e = spherical angular coordinate, rad K = property of the radiation source; ita dimensions depend on the source model X = wavelength, cm; also angle defined in Figure 11 p = absorption coefficient, cm-l v = frequency, s-1 = angle defined by eqs B.2 and B.3, rad p = spherical radial coordinate, cm a = distance defined by eq A.2, cm T = transmission coefficient, dimensionless $I = spherical angular coordinate, rad CP = overall quantum yield, mol einstein-' 7 = angle defined by eq 42.a, rad $ = angle defined by eq A.6, rad w1 = angle defined by eq 25, rad w2 = angle defined by eq 24, rad fl = solid angle, sr
-
u
= [(a + rI cos
For the variable 8:
For the variable 9: dD,(1,2)
= ?r +
'f
tc/
64.4)
where: (A.5)
tc/ = sin-'(rJu)
(A.6)
Appendix B Integration Limits for Indirect Radiation (Figure 4). For the variable p: PRf,(l,P)
=
-D cos t F [r; - D2 sin' 511/2 sin e
(B.1)
Here, B is the angle formed by the axial coordinate z and the direction of the reflected, incoming ray to the point of incidence I. In eq B.l we have sin 5 = -[(yp 1
D
+ k)cos dE - (xp + hMn dE]
(B.2)
+ k)sin #E - (xp + COS @E] (B.3) D D = [(yp + k)' + (xp + h)211'2 (B.4) yp = pI sin B cos d 03.5)
cos t = -[(yp 1
xp = pI sin e sin $I
(B.6)
Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 1341
-(h cos 4 + k sin 4) sin 6 [rW2- (h sin - k cos 4)2]1/2 03.7) sin 0 h = a + rI cos BI (B.8) k = rI sin PI (B.9) +
PI
=
= tan-
(1- m2)sin 4 - 2m cos 4
m=- YP + k xp h
+
]
(B.10) (B.ll)
For the variable 8:
with = pI sin 0 (B.13) For the variable 4: considering the rays coming from the regions of the reflector closer to the reactor tube (Figure 4b), the result is pi
4Rf,(l,2)=
+
(B.14)
and for those coming from regions of the reflector in the opposite position (Figure 4c), the result is (B.15) Literature Cited ALCOA, Aluminum Company of America. Technical Bulletin, 1964. Alfano, 0. M.; Casaano, A. E. Modeling of a Gas-Liquid Tank Photoreactor Irradiated from the Bottom. I. Theory. Znd. Eng. Chem. Res. 1988a,27,1087-1905. Alfano, 0. M.; Cassano, A. E. Modeling of a Gas-Liquid Tank Photoreactor Irradiated from the Bottom. 11. Experiments. Znd. Eng. Chem. Res. 1988b,27,1095-1103. Alfano, 0. M.; Romero, R. L.; Cassano, A. E. A Cylindrical Photoreactor Irradiated From the Bottom-I. Radiation Flux Density Generated by a Tubular Source and a Parabolic Reflector. Chem. Eng. Sci. 1985,40,2119-2127. Alfano, 0. M.; Vicente, M.; Esplugas, S.; Cassano, A. E. Radiation Field inside a Tubular Multilamp Reactor for Water and Wastewater Treatment. Znd. Eng. Chem. Res. 1990,29,1270-1278. Boyton, H. G.; Lewis, E. W.; Watson, A. T. A High Pressure Photochemical Reactor. Znd. Eng. Chem. 1959,51,267-270.
Cabrera, M. I.; Alfano, 0. M.; Cassano, A. E. Nonisothermal Photochlorination of Methyl Chloridein the Liquid Phase. AIChE J. 1991,37,1471-1484. Cassano, A. E.; Alfano, 0. M. Photoreactor Design. In Handbook of Heat and Mass Transfer;Cheremiainoff,N. P., Ed.; Gulf: Houston, TX, 1989;Vol. 3, Chapter 16,pp 583-670. Cassano, A. E.; Alfano, 0. M.; Romero, R. L. Photoreactor Engineering: Analysis and Design. In Concepts and Design of Chemical Reactors; Whitaker, S., Cassano, A. E., Eds.; Gordon and Breach Montreaux, Switzerland, 1986;Chapter 8, pp 339512. Cerdd, J.; Irazoqui, H. A.; Cassano, A. E. Radiation Fields Inside an Elliptical Photoreactor with a Sourceof Finite Spatial Dimensions. AIChE J. 1973,19,963-968. Cerdd, J.; Marchetti, J. L., Cassano, A. E. Radiation Efficiencies in Elliptical Photoreactors. Lat. Am. J. Heat Mass Transfer 1977, 1 , 33-63. Clarid, M. A.; Irazoqui, H. A.; Casaano, A. E. A priori Design of a Photoreactor for the Chlorination of Ethane. AZChE J. 1988,34, 366382. De Bernardez, E.; Cassano, A. E. A priori Design of a Continuous Annular Photochemical Reactor. Experimental Validation for Simple Reactions. J. Photochem. 1985,30,285-301. De Bernardez, E.; Clarid, M. A.; Cassano, A. E. Analysis and Design of Photoreactors. In Chemical Reaction and Reactor Engineering; Carberry, J., Varma, A., Eds.; Marcel Dekker: New York, 1986; Chapter 13,pp 839-921. Esplugas, S.; Vicente, M.; Alfano, 0. M.; Cassano, A. E. Effect of the Reflector Shape on the Performance of Multilamp Photoreactors Applied to Pollution Abatement. Znd. Eng. Chem. Res. 1990,29, 1283-1289. Gebhard, T. J. A study of Radiant Energy Distributions in PhotochemicalReactors and the Photochlorination of Trichloroethylene in a Stirred Vessel. Ph.D. Dissertation, Northwestern University, Evanston, IL, 1978. Holbrook, M. T.;Morris, T. E. Liquid Phase Chlorination of Chlorinated Methanes. US. Patent 4614572A, 1986. Irazoqui, H. A.; Cerdd, J.; Cassano, A. E. Radiation Profiles in an Empty Annular Photoreactor with a Source of Finite Spatial Dimensions. AZChE J. 1973,19,460-467. Jacob, S.M.; Dranoff, J. S. Light Intensity Profiles in an Elliptical Photoreactor. AIChE J. 1969,15,141-144. Jar&-M., E.; Galh, M. A. Refraction-Absorption Model of Light Intensity Distribution in a Tubular Flow Photoreactor. 1. Znd. Eng. Chem. Fundam. 1985,24,273-280. Maseini, J. J. Proc6d6de Fabrication de Chlorom6thanesSuperieura. E.P. 128818 Al, 1984. Vicente, M.; Alfano, 0. M.; Esplugas, S.; Casaano, A. E. Design and Experimental Verification of a Tubular Multilamp Reactor for Water and Wastewater Treatment. Znd. Eng. Chem. Res. 1990, 29,1278-1283.
Received for review September 17, 1992 Revised manuscript received March 2, 1993 Accepted March 5, 1993
