Ind. Eng. Chem. Res. 1993,32, 1342-1353
1342
The Multitubular Photoreactor. 2. Reactor Modeling and Experimental Verification Ana R. Tymoschuk,+Antonio C. Negro? Orlando M. Alfano,t and Albert0 E. Cassano'91 ZNTEC,I Guemes 3450, 3000 Santa Fe, Argentina
A single-lamp, multitubular photoreactor, surrounded by a cylindrical reflector, has been modeled and verified experimentally. The general case of a reactor having an absorption coefficient that depends on the reaction extent was considered, and the numerical results were compared with bench-scale experiments performed using the ferrioxalate photodecomposition reaction. The combination of a three-dimensional radiation model written in spherical coordinates with a threedimensional mass balance posed in cylindrical coordinates produced unusual numerical difficulties which made necessary the proposal of novel solution algorithms. Three possible simplifications of the model were also investigated. None of them produced good results. On the other hand, the rigorous model, not exempted from complexities, rendered excellent agreement with the experimental results. The analysis of the model and the experiments were performed with and without the presence of the cylindrical reflector.
I. Introduction The multitubular photoreactor consists of a tubular lamp placed at the center axis of the system, surrounded by several cylindrical tubular photoreactors; the system may be enclosed with a cylindrical reflector of circular cross section. (See De Bernardez et al. (19861, Figure 30, and Tymoschuk et al. (19931.) Multitubular continuous reactors are perhaps one of the most versatile types of photochemical systems as far as operating conditions and production rates are concerned. In fact, they can be operated under pressure without introducing serious complications in the possibilities of using heat and/or cooling devices. Similarly, they can be designed for an ample variety of production rates, or once constructed, they can be operated with a very flexible throughput. The lamp is separated from the reaction space facilitating assembly. Routine maintenance can be done also with simplicity and, with proper design, without interrupting the main operation. Such a device has not been previously modeled, and the only existing contribution in the pertinent literature is one work written by Tymoschuk et al. (1993). In this paper, the authors provided all the needed theory and an extensive set of numerical results for the modeling of the radiation field inside a multitubular photoreactor under the restricted conditions of negligible radiation absorption (diactinic reactors) or constant absorption coefficient systems (photosensitizedreactions). The main conclusions arrived at in the quoted paper are the following: (i) under normal conditions, there exist severe radial, axial, and angular nonuniformities in the radiation field inside the reactor, (ii) under normal conditions, the thermal energy and mass balances required for design most probably will have to be three-dimensional due to the important magnitude of the azimuthal asymmetries, and (iii) the contribution of the reflected radiation, even under the most favorable conditions, is not enough to compensate the asymmetries produced by the direct irradiation of the
* To whom correspondence should be addressed.
Assistant (UNL). Laboratory Technician (CONICET). $ CONICET Research Staff Member and Professor at UNL. A Instituto de Desarrollo Tecnol6gico para la Industria Q u h ica. Consejo Nacional de Investigaciones Cientificaa y TBcnicas (CONICET) and Universidad Nacional del Litoral (UNL). + Research
t
reactor tubes; moreover, with proper design of the whole system the contribution of the reflected radiation should be unimportant always. All the information reported in that paper, from now on paper 1,is substantial to this one and we will refer often to it. In this work we report the modeling of a multitubular photoreactor operating under conditions of having a variable absorption coefficient as a consequence of the existence of changes in concentration produced by the chemical reaction. The system is analyzed under isothermal conditions, and the mass balance, for a simple reaction, is developed by using a three-dimensional model and solved numerically. Predictions obtained with the numerical solution of the conservation equations (mass and radiation balances) and information regarding a wellknown reaction intrinsic kinetics corresponding to an actinometric reaction are compared with experimental results obtained with the multitubular system working with three different reactor tube radii. One rigorous model and three approximations (a complete absorption of the incoming radiation model, a constant absorption coefficient model, and a two-dimensional mass balance with variable absorption coefficient model) are compared with the experimental results obtained by using the potassium ferrioxalate decomposition reaction. None of the models make use of experimentally adjustable parameters. With the obtained results, one can reach definitive conclusionsabout the most convenient procedure for reactor design, from both the quality of the results and the complexity of the model points of view. 11. Theory
1. The Reaction. Stoichiometry and Reaction Kinetics. The reaction chosen to test the model is the well-known actinometric solution of potassium ferrioxalate (Parker, 1953; Hatchard and Parker, 1956; Parker and Hatchard, 1959). The experimental details are reported in section I11 (Experiments). The global reaction, for which the radiation wavelengthdependent overall quantum yield is known, is
hu
2Fe3++ C,O;2Fe2++ 2C0, (1) The reaction kinetics has been studied and reported following the formation of the ferrous ion, and it can be
QSSS-5SS5/93/2632-1342$O4.00/0 0 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 7,1993 1343 expressed in terms of the overall quantum yield and the local volumetric rate of energy absorption (LVREA). In its turn the LVREA is a function of the incident radiation and the absorption coefficient. The absorption coefficient cannot be expressed as a simple function of the reactant concentration because the reaction product also absorbs. It must be split into two parts: one corresponding to the photochemicallysignificant reactant absorption, which has kinetic and attenuation effects, and a second corresponding to the inner filtering effect produced by the product, which contributes only to attenuation. Only the first one intervenes in the reaction kinetic expression explicitly. Thus, we have R = ~ v z @ , edv sy y1
where R is the reaction rate, is the overall quantum yield, eayis the local volumetric rate of energy absorption, pr,,,is the reactant absorption coefficient, G, is the incident radiation, ar," is the reactant molar absorptivity, and C, is the radiation absorbing reactant concentration. Hence, through this expression for the reaction kinetics, and particularly because of the relationship between radiation absorption and concentrations, the radiation and the mass conservation equations are coupled. If almost monochromatic radiation is used, the integration over the frequency interval is not required. The specific form of the absorption coefficient in eq 4 will be given in section 111. 2. Conservation Equations. Radiation Balance. The incident radiation (G,in eqs 3 and 4) must be known at every point inside the reactor. Its value is related to the emission power of the lamp, the geometric and optical characteristics of the reactor and the reflector, and the absorption of radiation by reactants, inerts, and products along the radiation path inside the reactor. Due to the chemical reaction, concentrations inside the reactor are not uniform, and consequently, the rigorous model, even under steady-state conditions, must include a positiondependent, variable radiation absorption coefficient. In paper 1, the radiation field inside a multitubular photoreactor was studied. With this purpose the radiative transfer equations for this type of reactor were presented in detail and the reader is referred to paper 1. Basically, a radiation emission model for the source, a radiation reflection model for the reflector, and a radiation absorption model for the reactor are linked by systematic application of the radiative transfer equation. Contributions from the lamp and from the reflector are added to obtain the incident radiation at any point within the reaction space. For direct and indirect (reflected) radiation the results are
H(B,6) = exp ( - J PPO p dp*)
(7)
Expressions for the functions FDand Fw,as well as the limits of integration for eqs 5 and 6, can be found in paper 1. Since now the absorption coefficient is a function of position, eq 7 is more difficult to solve. Details related to the calculation of the function H(B,4) for the case of P variable with concentration will be given in part 3 of this section.
Mass Conservation Equation. Since the reaction is carried out using a dilute actinometric solution and the heat of reaction is not significant, the system can be maintained under isothermal conditions with no difficulties. In addition, thermal effects on the kinetics of a photochemical reaction are not very significant in the majority of the cases. Both situations together make unnecessary the inclusion of the thermal energy balance for the design. I t should be noted that this approximation is valid when both conditions are fulfilled and, at the same time, the heat produced by the radiation source is not very significant. For the reaction described by eq 1, the mass balance for a single component is sufficient; thus, the product formation (Fe2+)will be used as the key component. The following assumptions are made: (i) steady state, (ii) laminar flow of a Newtonian fluid, (iii) constant physicochemical properties, (iv) axial diffusion negligible as compared with convective flow in the same direction, and (v) isothermal behavior. The mass balance for the key component, the ferrous ion, in dimensionless form results:
The boundary conditions are for 0 Iq I1 and 0 I w I 2, at y = 0:
+ = finite
for 0 I q I 1 and 0 I w I 2, at y = 1:
d+/dy = 0 (8.b)
for 0 Iq I1and 0 I y I 1, at w = 0:
a+/&
for 0 Iq I1 and 0 I y I 1, at w = 1:
a+/dw = 0 (8.d)
(8.a)
= 0 (8.c)
and the initial condition is
+
for 0 I y I 1 a n d 0 I w 5 2 , a t q = 0: =0 The dimensionless reaction rate results:
(84
The product output averaged concentration for each reactor tube is obtained by means of: ( + ( y , ~ ) ) ~= =1 4 ~ J o 1 + ( ~ , w ) C 1-
r2)dy d w
(10)
3. Numerical Solution. Equation 8, with boundary and initial conditions (8.a-8.e) is a three-dimensional, partial, integro-differential equation that can be solved by using some of the known variations of the finite difference techniques. Initially, eq 8 is rendered discrete using uniform intervals in the three cylindrical coordinates (axial, radial, and angular). The resulting system is solved by using the alternate direction, implicit method (ADI), often used to tackle this type of problem (Peaceman, 1977). To start the numerical solution, the system is solved assuming first a constant absorption coefficient. This solution, which is easier to obtain (see paper 11, produces a good initialization for the solution of the more exact problem formulation (the one with an absorption coefficient which is a function of the reaction extent). As can be readily deduced from the results of paper 1, the incident radiation for systems of high absorption coefficient has a predominant contribution in the region of the reactor which receives direct radiation (vicinity of w = 1). At points not too separated from the reactor wall (y = l), the reaction is completed and the reactant is depleted totally. Conversion of the reactant is less significant in other points of the reactor, particularly in
1344 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993
the vicinity of w = 0. The use of an actinometric reaction, which has generally strong radiation absorption characteristics, falls under these characteristics. Consequently, the number of grid intervals in the radial direction is very important; if the number of intervals in the radial mesh is too small, much of the required precise information representing the concentration profiles in the regions of maximum conversion is lost, particularly because of the influence of the wall boundary condition (null mass flux). On the other hand, if the number of mesh points is too large, the computer processing time, particularly in the case of having a variable absorption coefficient, becomes too large. The physical situation occurring at the region of the reactor wall closer to the lamp is similar to the steep changesin concentration occurring inside a boundary layer. Problems of this sort are satisfactorily solved by using unequally spaced grid points for the integration mesh in the conflictive direction or, alternatively, some equivalent effect, by an appropriate transformation in the radial variable. Thus, one looks for some transformation in the radial variable that tends to accumulate the grid points in the region where most of the changes occur, and have them more separated in those regions where the changes are smoothest. One such a technique transforms the variable under consideration in the differential equation and carries on the transformation to the equations written in a discrete manner. Afterward, the same numerical method is used with a grid of uniform separation, but using the equations written in the transformed space. This type of transformation is 2-fold convenient: (i) it solves the problem described above and (ii) it maintains the uniform separation in the actual integration grid, which will be very important for the solution of the problems that will be described below. Details of the radial transformation are included in Appendix A. The fact that in eq 7 the absorption coefficient p is a function of the reactant and product concentrations gives rise to additional computational difficulties. The computation of the integral in eq 7 must be performed along the radiation ray trajectory (which corresponds to the p coordinate of a spherical coordinate system). The mesh in which the mass conservation equation has been written corresponds to a three-dimensional cylindrical coordinate system. Essentially we have
We have assumed that we can express the absorption coefficientas a function of the product concentration alone, and that the product concentration can be obtained by integration of the mass balance, integro-differential equation at each point of the finite difference mesh. Expressing the absorption coefficient as a function of a single component is not always possible, but, if needed, the use of a more complex function will overcome the difficulty adding only minor conceptual complications in this particular aspect of the problem at stake. The second computational problem is generated by the three-dimensional, spherical characteristics of the radiation propagation. The trajectory of a ray defined by its direction in the p, 6, @ coordinate system does not necessarily coincide (in fact, it coincides only by chance) with the location of the points of the three-dimensional integration mesh defined in the cylindrical coordinate system used for the mass balance (y, 7 , w ) . In order to compute the attenuation of each ray along its trajectory, we are required to know the encountered concentrations.
To do it, in general, one needs to interpolate between known values of concentrations in the vicinity of the point under consideration. These interpolations will have to be made, in general, for two of the three dimensions of the problem. The computational procedure is described in Appendix B. The third computational difficulty arises because the three-dimensional problem does not have, for all angular directions, a condition of concentration symmetry for the radial coordinate a t the reactor center. Thus, the concentration at the center of the reactor must be calculated; the boundary condition 8.a indicates that at the reactor center the concentration must be finite. Under this limitation, the differential equation written in cylindrical coordinates cannot be used to evaluate the concentration at y = 0. Details of the concentration calculation at the reactor center are included in Appendix C. The last problem, which is originated in the integrodifferential nature of the model mathematical structure, can be described as follows. At any point I inside the reactor one needs to compute the rate of reaction. According to eq 3, -93 depends on the local value of pt (which is a function of the local concentrations at the point I) and on G. The value of G depends on the history of every one of the radiation rays that arrive at point I from all the directions associated with the characteristics of the lamp emission and the reflections from the refledor; Le., starting from an emission point in the lamp, each ray, in a direct or reflected manner, reaches the reactor and travels from the point of entrance at the reactor wall to point I. Along the complete ray trajectory, a ray suffers attenuation of two types: (i) when it is reflected, by absorption in the reflector (which is accounted for by means of the reflection Coefficient), and (ii) when it enters the reactor, by absorption inside the reactor until the ray reaches point I. At every point inside the reactor, there exist different species concentrations (which depend on the reaction rate at those points) which produce different values of the absorption coefficient. Hence, the attenuation by absorption of each ray along its path depends on the species concentrations existing a t all the points that the ray transverses or, which is the same, depends on the value taken by the reaction rate a t all those points. In other words, the evaluation of the reaction rate at any one point demands the knowledge of the reaction rates at almost all other points inside the reactor. Consequently, an iterative procedure must be superimposed on the finite difference solution, looking for convergency in the concentration mesh of the whole reactor, simultaneously. Starting from initial values of concentrations for the whole reactor, one must solve the problem until essentially “steady-state” concentrations of the product are achieved. This is done by sweeping the finite difference mesh as many times as needed until an acceptable, prescribed, minimum error is attained (see Romero et al. (1983) and Cassano et al. (1986)). Once the numercial procedure have been implemented considering all the solution algorithms previously described, one can proceed with the application of the resulting computational program. The problem was solved in an old Digital-Vax 11/780, operated on a time-sharing basis, with a VMS operating system and having 4 MB of core memory. Usually the transformation parameter (m) was set equal to 2. The three-dimensional mesh had 21 radial points, 16 axial points, and 19 angular positions. With variable attenuation (depending on the steady-state point concentrations of the full integration region) the complete solution of the three-dimensional problem required, after proper initialization, sweepingthe whole finite different mesh with a maximum of five iterations; the
Ind. Eng. Chem. Res., Vol. 32, No. 7,1993 1345 Table I. Summary Description of the Reactor Part reactor tube (quartz)
reflector (aluminum,with Alzac treatment) germicidal lamp (Iwasaki, 1975)
Table 11. Summary Description of the Reaction Conditions parameter units quantum yield (254 nm) mol/einstein molar absorptivity, ( ~ ~ ( 2 nm) 54 cm2/mol diffusivity cm2/s K operating temperature init concn of potassium ferrioxalate mol/L
parameter length radiw distance a transmissioncoeff length radius reflection coeff (av) nominal input power nominal output power main wavelength equiv photochem power (254 nm)
value 1.2 4.73 x 1 P
1x10-6 293 6x104
Expressed in the log basis. Initial value of pr (cm-l) = 2.3ar(6 x 10-6).
V
Figure 1. Experimental apparatus: A, B, C, D, polypropylenetanke, E, centrifugal pump; F, starter; G, quartz reactors; H, UV lamp; J, lamp operation meter (V, A, and W);K, circular reflector; P, Q, sampling valves; R, ballast.
employed CPU time, including appropriate initialization of the initial conditions, was 390 min. The case of a constant absorptioncoefficient was solved, under the same conditions, in 40 min. Tables I and I1 indicate the values of the parameters employed in the numerical solution of the mathematical problem.
111. Experiments Equipment Setup. Figure 1 shows a schematic flow sheet of the experimentalapparatus. All experimentswere conducted with a single tube. Three tubes of different radii were used. The system also allows for variations in the distance from the reactor to the lamp (a). The reactor was enclosed alternatively by a reflector of circular cross section or by a nonreflecting surface of the same shape. The reactor was made of quartz, Suprasil quality. The reflector was made of aluminum, specularly finished, with Alzac treatment (ALCOA, 1964). Precise positioning devices were used to locate the lamp at the center of the system and each one of the reactor tubes a t the desired distance from the lamp. Details of the lamp, the reactor, the reflector, and their dimensions are provided in Table I. The lamp was chosen with an emission which is almost monochromatic to simplify computations. A centrifugal pump made of glass and Teflon was used to circulate the reactants. At the inlet and outlet sections of the reactor tube, samplingdevices allowed the collection of the ingoing and outgoing solution for analysis. A precise controlling device of the operating power, operating voltage, and operating current was installed because we found that experimentalconversions were not reproducible
units
value
m m m
0.60 0.002-0.005-0.010
-
0.050.060.065
m m
0.60 0.08 0.65 30 8.3 254 1.765 X 1V
-
W W nm einstein/a
1
when the lamp was not operated at the rated conditions. The reaction temperature was kept constant by proper conditioning of the laboratory; this was possible because the heat generated by the lamp was negligible. Four 200-L polypropylene tanks completed the setup. One contained the reactant, the second collected the reaction product, the third had distilled, demineralized water for cleaning procedures after the shutdown of an experimental run, and the fourth was used to collect any sort of wastes coming out of the reactor during cleaning procedures. The whole system, with the exception of the illuminated length of the reactor, was protected from laboratory light. Procedure. The reacting solution of potassium ferrioxalate was prepared according to the description provided by Murov (1973) from a solution of potassium oxalate and a solution of ferric sulfate. The solution must be kept at pH = 2.5-3.0~iththe addition of sulfuric acid. The desired mole ratio must be of about 6 mol of oxalic acid for each mole of ferric sulfate. The reaction was not used with actinometric purposes and, consequently, in order to facilitatethe penetration of the radiation inside the reactor, the minimum recommended molar concentration of potassium ferrioxalate was employed (0.006 M). The ferric sulfate concentration was standardized with a solution of ethylenediaminetetraacetic acid (EDTA), buffered with glycine and using salicylic acid as indicator. The ferrous ion formed during the photoreaction was subsequently determined via spectrophotometric determination of its phenanthroline complex, buffered with sodium acetate, at 510 nm using a Cary spectrophotometer. The phenanthroline complex was standardized with standard ferrous sulfate solutions in 0.1 M sulfuric acid; they were titrated with potassium chromate solution using diphenylamine as indicator. Once prepared, the solution was fed to the reactor. After steady-state conditions, measured in terms of flow rates and output concentrations, were reached, the lamp was turned on. The system was operated during a period of about 120 min to wait for the steady operation of the lamp (toreachthe rated conditions of power, voltage, and current input), constant flow rate, and constant temperature. Finally, samples were taken every 15 min to check for the steady-state operation of the reactor (by proper measurement of the reactant conversion). Once this routine was completed, a new flow rate was adjusted to operate the reactor under a different output concentration. By repetition of this procedure, several experimental points were obtained which allowed the construction of the required plot of ferrous ion concentration vs mean residence time. It was found that in using the germicidal lamp the presence of the ferrous ion acta as an internal filter; i.e., the ferrous salt also absorbs (Parker, 1953). In order to properly account for the absorption by the reactants and the product, the absorption coefficient of the reacting mixture was obtained as a function of the ferrous ion concentration. The experimental results, obtained spec-
1346 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993 100
r
251 0 0.00
I
,
,
0.25
0 50
+
;,1 %.
0.75
1.00
Figure 2. Total absorption coefficient as a function of the dimensionless product concentration.
trophotometrically for a wavelength equal to 254 nm, are shown in Figure 2. The different samples were obtained by irradiating the mixture of potassium ferrioxalate in a well-stirred batch photoreactor. By linear regression analysis the following equation represents the absorption coefficient of the 0.006 M potassium ferrioxalate initial concentration: p = 67.31 - 61.00# (13) Equation 13 shows that even when the potassium ferrioxalate concentration is totally depleted (# = 11,there persists a residual absorption produced by the reaction product. This value of the absorption coefficient must be used to compute the attenuation of radiation (eq 7). However, the rate of reaction is proportional only to the absorption produced by the potassium ferrioxalate alone; hence, in eq 13, the inner filtering effect must be excluded and the equation to be used for the kinetically useful absorption coefficient in eq 4 is p., = 67.31(1- #)
(14)
IV. Results The results will be analyzed using four different models of increasing degree of complexity. The numerical values will be compared with the experimental results. The experimental procedure measured the ferrous ion concentration by means of the phenanthroline complex at the outlet of the reactor tube. Hence, the mixed cup output concentration of the product was the experimental value corresponding to the dependent variable. The flow rate, the reactor radius, and the distance from the reactor to the lamp were the experimental independent variables. Thus, we will have, in general, plots of the experimental values of Fez+ concentration vs the mean residence time inside the reactor (7).These values were obtained always with a single tube (but using alternatively three different reactor radii). In the majority of the runs, a well-designed multitubular reactor was simulated by eliminating all forms of reflection; let us recall that, in general, a welldesigned photoreactor should leave just the minimum required spacing between reactor tubes, and each one of the tubes should absorb as much of the available radiation as possible; if this is the case, not much chance for significant contributions of the reflector remains. In a second set of runs, the reflector was included, but then the used computed results were obtained for a single tube with all forms of reflections taken into account (see paper 1, where the problem was stated and solved, but for a constant radiation absorption coefficient). The four models are as follows:
Model 1: All the available radiation reaching the reactor is absorbed and produces the reaction product. The reactor operates as a blackbody. Then, the radiation model must provide the value of Q expressed in einsteins s-l by integrating over the reactor surface all the incoming radiation energy fluxes. (See Cerdd et al. (1977) and eqs 29,31,34, and 36 of paper 1.) Then, a macroscopic mass balance in the reactor gives F [ ( C ~ e z + ) o u t- ( c F & + ) h ]
= @Q
(15)
The inlet ferrous ion concentration is 0. The flow rate F may be substituted by the ratio of Vd7. Thus, the predicted values of the product output concentration can be calculated from eq 15 because the overall quantum yield, the flow rate, and the reactor volume are data and Q is provided by the model. To compute Q, only the reactor, reflector, and lamp specifications are required. These are results of a blackbody reactor with a threedimensional radiation model and a macroscopic mass balance. Model 2 The absorption coefficient is constant (independent of the changes in the species concentrations). Equation 7 can be readily integrated to obtain
H(4d = exp(-pAp) (16) and Ap is obtained with eqs 12,16, and 17 of paper 1. In this case, all the difficulties associated with the concentration dependence of p are bypassed and, consequently, no use of eqs B.l-B.23 is required. The solution of eqs 8-10 provides the output concentration of the ferrous ion. The Pe and Ge numbers may be calculated from the reactor specifications, the physicochemical properties of the system, and the flow rates. The mass balance requires the known overall quantum yield, the reactant inlet concentration, and the reactor and lamp specifications. These are the results of a constant absorption coefficient reactor with a three-dimensional radiation balance and a three-dimensional mass balance. Model 3 The unavoidable three-dimensional characteristic of the radiation field is maintained, but a twodimensional mass balance is used. The absorption coefficient is a function of the species concentrations. The problem must be solved for selected values of the angular coordinate p. To provide the numerical solution, eq 8, modified to disregard angular diffusion, is used (mathematically it is equivalent to assume azimuthal symmetry for concentrations); boundary conditions 8.c and 8.d are not needed. Since the three-dimensional characteristics of the radiation propagation are kept, the contributions to the incident radiation (G) from the whole threedimensional space to a two-dimensional mass balance grid must be incorporated. To do it, without loss of rigor, the problem can be solved by using a cylindrical projection of all concentration values on the rectangular mesh used in the mass balance numerical solution (a mesh located in a selected angular position). Even though the model considers that the absorption varies with the changes in concentration, it must be noted that the cylindrical projections mentioned above are made on the basis of a model that computes the species concentrations assuming azimuthal symmetry in the mass balance written for a selected angular position. The cylindrical projection approach was initially suggested by Romero et al. (1983) for an annular reactor and successfully used by Claril et al. (1988) and Cabrera et al. (1991a-c) in cylindrical reactors with direct and reflected radiation. The reader is referred to the original works for the details. The required information is the same as the one used in model 2. The product output average concentration is obtained
Ind. Eng. Chem. Res., Vol. 32, No. 7,1993 1347 I
61
2
0
(a)
a/3
2a/3
10 I
T
I
(b)
Figure 3. Comparison of 2-D and 3-D mass balance models. (a) Model 3 (- - -) and model 4 (-) predictions versus experimental data (angular position of the 2-D mass balance is the parameter). (b) Model 3 (- - -) predictions versus model 4 (-) predictions (mean residence time is the parameter).
using eq 10 with +being a function of y only. As a matter of fact, computational results will produce different output concentrations for different /3 positions. These are the results of a three-dimensional radiation model with a two-dimensional mass balance. Due to the already known severe angular asymmetries of the multitubular photoreactor, one should expect that the results of this model will be very much dependent of the /3 position chosen for performing the numerical solution; moreover, the model performance should be very poor when the reacting solution is optically thick. Model 4: This is the three-dimensional radiation, threedimensional mass balance model, with variable absorption coefficient, presented in section I1 of this paper. Since model 3 will produce different outputs according to the chosen angle, to facilitate the graphical presentation of the results, first, we will compare this model with the experimental results and with model4. Afterward, results from models 1,2, and 4 will be presented simultaneously with the experimental data, under a wider range of operating conditions. Figure 3a is a plot of the mixed cup product output concentration as a function of the mean residence time. These results correspond to a reactor without reflector, having r R = 0.5 cm and a = 5 cm. The plot shows the experimental data; results from model 4 are presented with solid lines and broken lines represent those results produced by model 3 for values of 6 equal to T,2 d 3 , and s / 2 . Predictions for /3 = 0 give extremely low conversions; hence these results fall on a line that almost coincide with the abscissa. Clearly, it seems that, for this set of operating 2 equivalently conditions, an angular position close to ~ / (or 3 ~ 1 2could ) be representative of the reactor actual behavior; the error is in the order of 25 % . In fact, this agreement isjust meaningless,because it is only valid for the particular values of Q, a, pr,and p existing in this run. Any change in these parameters or variables may render a different "representative" angular position for the reactor performance. In order to observe more effectively the effects produced by ignoring the angular asymmetries, let us look at Figure 3b, where model 3 is compared with model 4 under the same conditions. Now T is the parameter and the mixed cup output concentration is plotted against the angular position. The results for model 4 are independent of the angle. It can be seen that, for these particular runs, for angles in the neighborhood of 7r/2 all the results of model 3 coincide with those obtained from model 4. Thus, from the maximum value of the computer results (when the whole reactor performance is assumed to be repre-
0
20
40
60
(C)
T(S) Figure 4. Results for models 1,2, and 4 versus experimental data. (-) Model 4; (- -) model 2; (- - -) model 1. (a) Q = 0.2 cm; (b) Q = 0.5 cm; (c) Q = 1.0 cm.
-
sented by a /3 position equal to T)to the minimum value (/3 = 0), there will be always some chosen angular position such that a two-dimensional mass conservation model can represent the reactor actual operation; but it is impossible to anticipate it without a three-dimensional mass balance model or the experimental data. We conclude that, for this system with large values of p, it is impossible to use Model 3 for performance predictions and/or scale-up purposes. Figure 4 depicts the comparison of models 1, 2, and 4 with the experimental data, for three different reactor radii, The value of a is equal to 5 cm, and these results were obtained for a system without a reflector. There are no doubts that model 4 represents very well the actual reactor performance: deviations from the experimental data are never greater than 4 5%. Under our operating conditions, model 2 underestimates the production rate. It would be possible to choose a value of the constant absorption coefficient, different than the one employed in these runs (the initial conditions), that will fit well the data. However, again it will depend on the operating conditions, and consequently, model 2 has the same drawbacks that we found in model 3. It is interesting to note that, for certain values of the operating variables, and choosing a lower value for the constant absorption coefficient than the one corresponding to the initial conditions, model 3 could even overestimate the reactor production. This is typical of oversimplified photoreactor models; they cannot properly take into account the unusual effects produced in photoreactor performance when the
1348 Ind.
Eng. Chem. Res., Vol. 32, No. 7, 1993 'O
I t
t
0
0
0
20
40
60
T(S) Figure 5. Results for model 4 versus experimental data. (-) Model predictions. Experimental data: (0)with reflector; (0)without reflector.
photochemical optical density is changed (we are defining here the concept of optical density as the product of the absorption coefficient times the radiation propagation characteristic dimension, in our case the reactor radius). Model 1is a useful tool to predict, in a simpler manner, the maximum possible reactor performance. Due to its relative simplicity, it can be used for preliminary calculations and, more specifically, to compare different multitubular photoreactor arrangement performances (see paper 1). However, for large mean residence times, this simple model will lead to predictions of these maxima with excessive error. A very important effect is observed when one moves from Figure 4a to 4b and 4c. As the reactor radius increases, the conversion decreases significantly. This is a typical phenomenon of photoreactors having a reacting medium which is optically thick (highly radiation absorbing). This effect is very important when conversion and production rates must be optimized. This very unusual behavior in conventional, tubular reactors is often ignored in photoreactors when the proposed models do not take into account the reactor radius as one of the important characteristic dimensions. Two typical examples are (i) one-dimensional models where only the axial dimension is considered and (ii) Two-dimensional models rendered unidimensional by performing an average procedure of the radiation field over the cross-sectional area. Figure 5 is the confirmation of the abilities of model 4 to predict the performance of the multitubular photoreactor under any experimental conditions. For a reactor with 121 = 0.5 cm and a = 5 cm the squares represent experimental data obtained without the reflector; the solid line represents predictions from model 4. The circles represent experimental values obtained with a single tube with the reflector, and the solid line again represents predictions produced by model 4 with the inclusion of all the contributions produced by reflected radiation. The agreement is very good. It must be mentioned that these last results represent the maximum performance of the reflector, because only one tube is operating in the system. One should remark once more that a properly designed multitubular photoreactor will have as many reactor tubes as possible and the contributions from reflected radiation will be greatly reduced. I t has been indicated (paper 1) that one should never expect an increase in performance much greater than 20-25 5% when the reflector is added to a well-designed multitubular reactor. Figure 6 shows experimental and computed predictions for the performance of the system when the distance from the lamp to the reactor is changed. The first one is for r R = 0.5 cm and the second is for 121 = 1 cm. In both cases 7 is the parameter. For comparison purposes data computed with model 1are also included. Once more, the
5
7-
a(cm)
0
I=-------7.23 s
7
5
a(crn)
(a1 (b) Figure 6. Results of models 1 and 4 versus experimentaldata. Effect of the distance a on product conversion. (- - -) Model 1; (-) model 4. (a) n = 0.5 cm; (b) = 1.0 em.
agreement between the results produced by model 4 and the experimental data is excellent. I t can be concluded that, whenever possible, the distance from the lamp to the reactor must be shortened. In this figure it can be observed that predictions from model 1 are in better agreement with the experimental data when (i) the mean residence time is small or (ii) the separation of the lamp from the reactor is large. I t seems appropriate at this point to stress that model 4 does not make use of any experimentally adjustable parameter, and consequently, within an ample range of design parameters, it can be used for design or scale-up purposes with a high degree of reliability.
V. Conclusions A rigorous model for a single-lamp, multitubular photoreactor, with and without reflector, has been developed. By comparingthe results with experimental data in a bench scale apparatus, the following can be concluded: 1. The modeling of this type of photoreactor requires the use of a three-dimensional radiation balance and a three-dimensional mass balance. When the absorption coefficient is a function of the species concentration, the numerical solution is difficult and the computer processing time is greatly increased. The agreement between the proposed model predictions and the experimental data is excellent. Under the studied conditions none of the possible simplifications rendered acceptable predictions when they were compared with experimental data. 2. A simpler model, called a blackbody reactor, can be used to predict the maximum possible performance of the multitubular photoreactor. However, even these predictions would be useful only for short mean residence times. 3. Only for the case of a truly photosensitized reaction, where the absorption coefficient is independent of the reaction extent, a third model, maintaining the representation of the three-dimensional nature of both the radiation and the concentration fields, can be used. In this case, the computer time is greatly reduced. 4. Under no circumstances can a model that reduces the mass balance to a two-dimensionalformulation be used. Its results have a wide range of variations depending upon the angular position that is chosen for performing computations. The only possible exception, and only as an approximation, would be a system having very low radiation absorption. 5. Special numerical procedures were devised in order to solve some of the computational difficulties associated with the rigorous model. Among them, one can remark (i) a coordinate transformation employed to solve the case of a highly radiation-absorbing reacting medium, (ii) the superposition of a three-dimensional radiation balance
Ind. Eng. Chem. Res., Vol. 32, No. 7,1993 1349 written in spherical coordinates with a three-dimensional mass balance expressed in cylindrical coordinates, and (iii) the evaluation of the three-dimensional radiation path inside the reactor. 6. In tubular flow reactors (i.e., reactors where there is not perfect mixing in the radial direction) increasing the reactor radius may decrease the output conversion. In fact this phenomenon is always observed when the reacting medium is optically thick. 7. The proper design of the single-lamp, multitubular reactor generates a complex optimization problem. Increasingthe distance from the lamp to the reactor decreases conversion but gives more room to include more reactor tubes. Thus, total throughput and total conversion are in conflict. At the same time, the value of the radiation absorption coefficient controls the optimum reactor tube radius; this control may impose restrictions on the radius of the circumference defined by the lampreactor distance. This is a unique feature of photochemical reactors generated by the characteristics of the radiation transport.
The authors are grateful to Consejo Nacional de Investigaciones Cientificas y TBcnicas (CONICET) and Universidad Nacional del Litoral (UNL) for financial aid.
T = transmission coefficient, dimensionless 4 = angular spherical coordinate, rad @ = overall quantum yield, mol einstein-1 $ = CIC”, dimensionless concentration w = @IT, angular cylindrical coordinate, dimensionless D = rate of reaction, dimensionless Subscripts D = direct incident radiation I = incident point at the reactor or a property of an incident ray L = lamp property r = reactant property R = reactor property Rf = indirect incident radiation v = frequency o = relative to the surface of radiation entrance Superscripts a = absorption = value projected on the x-y plane - = average value, used in eqs B.12, B.18, B.21, C.2, and (2.3 O = inlet value * = relative to the attenuation trajectory Special Symbols ( ) = average value, used in eqs 9 and 10
Nomenclature a = distance between lamp axis and reactor axis, cm C = concentration, mol cm-3 D = diffusivity coefficient, mol cm-2 ea = local volumetric rate of energy absorption,einsteins cm-3
Appendix A Transformation in the Radial Variable. Following the suggestions of Kalnay de Rivas (1972) and particularly Agrawal and Peckover (1980),several transformations were investigated. One which produced satisfactory results is
Acknowledgment
5-1
E = lamp output photonic power, einsteins s-l F = volumetricflow rate, cm3s-1; also function defined by eqs 4.a and 7.a of Paper 1, cm G = incident radiation, einsteins cm-2 s-1 Ge = ~ L Rgeometric , number, dimensionless h = Planck’s constant, J s H = function defined by eq 7, dimensionless L = length, cm m = parameter defined by eq A.2, dimensionless N = total number of mesh points, dimensionless Pe = 2 d D ( u ) , Peclet number, dimensionless Q = total incident radiation, einsteins s-1 r = radius or radial coordinate, cm 3 = reaction rate, mol cm-3 s-1 T = coordinate along the radiation ray defined by eq B.l, dimensionless u = velocity, cm s-1 V = volume, cm3 x = rectangular coordinate, cm y = rectangular coordinate, cm Greek Letters a = molar absorptivity, cm2 mol-’ j3 = angular cylindrical coordinate, rad y = rim, radial cylindrical coordinate, dimensionless r = reflection Coefficient, dimensionless 6 = “boundary layer” thickness, dimensionless 7 = zILR, axial cylindrical coordinate, dimensionless B = angular spherical coordinate, rad t9 = function defined by eq B.19, dimensionless K = E L J ~ T ~ property ~ ~ ~ L ofL the , radiation source, einsteins cm-3 s-1 s r l p = absorption coefficient, cm-1 v = frequency, 5-1 4 = transformed radial cylindrical coordinate,dimensionless p = radial spherical coordinate, cm a = distance defined by eq A.2 of paper 1,cm T = residence time, s
The parameter m, included in the hyperbolic transformation, is a number, not necessarily an integer, which can be obtained from the following equation: m = tanh-’[(l- S)’/2] (A.2) where d is the boundary layer thickness. The value of m is related to the necessary number of grid points (n),out of the total number of points used in the direction of the mesh under consideration (N), that must be included in this “boundary layer” to obtain fast and sure convergency. In a photoreactor, one can define the thickness of this layer by taking the point where the incident radiation has changed 99% from its original value (i.e., the value of the incident radiation is 0.01%I of the one arriving at the reactor wall). In our system, the most critical situation is found at a point which is located in the vicinity of w = 1(B = T ) and it is not too separated from y = 1. In order to speed up the determination of the parameter m, one can resort to the use of an approximate model. With this purpose, the linear source with emission in parallel planes (LSPP) model was used (Harris and Dranoff, 1965; Cassano et al., 1986). With this limited purpose, and for the cases in which the “boundary layer” is thin (high radiation absorption), the approximation should produce good results. According to this model, the incident radiation is given by (A.3) where a is the distance from the lamp center line to any point inside the reactor space and is the distance from the lamp center line to the point of incidence a t the reactor wall. An expression for B can be found in paper 1,eq A.2.
1350 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993
0'08
7
2
0
m
6
4
Figure 7. Selection of the parameter m for the coordinate transformation. Table 111. Estimation of the Parameter m I.L (cm-9 a (cm) ly( (cm) 65
0.2 0.5 1.0
6.5
6
m
0.35 0.14 0.07
1.1 1.6 2.0
Considering our definition of the boundary layer, the dimensionless thickness of fluid for an angular position fixed at /3 = A, results: 6=--
a-a , rR
ln[ O.Ol( 1 +])&6
--
Yn
m-1
Yn+1
Figure 8. Computation of the attenuation of a radiationray which propagates with spherical emission (characterized by p,B,+) using data of the mass balance cylindrical integration mesh (y,q,@).
(A.4)
TRP
Note that eq A.4 indicates that 6 is a function of P,a, and This is important because a different value of m may be required for each reactor diameter, each species radiation absorption coefficient, and, to a much lesser extent, to each distance from the lamp to the reactor tube; the second in its turn, depends upon the concentration of the said species. However, one can always choose conditions where the problem is more severe and use the resulting value for the rest of them. Table I11 shows the calculated values of m for different reactor radii. These results were obtained for a distance of 6.5cm between lamp axis and reactor axis and a constant absorption coefficient of 65 cm-l. First, eq A.1 was used to transform eqs 8 and 8.a-8.e. The integro-differential equation was solved using eqs 5-7 but with p = constant (in this first approximation), calculating the product output average concentration with eq 10. The result was plotted as a function of m starting with a value of m = 1 as depited in Figure 7. It is clear that with N7 = 21, NV = 16, and N , = 19 a value of m between 2 and 3 is satisfactory. It should be noted that, with the same number of grid points and using the nontransformed equations, the product output average concentration obtained after running the program during the same computer time differed from the correctly converged result by almost 1 order of magnitude. rR.
'
"A
Figure 9. Attenuation path for any position I and for any (e,+) ray.
center at I),such that their corresponding radial cylindrical coordinate (y)coincides with a point falling on the radial integration mesh (Y,,); in this case we will say that the averageprocedure will be pivoting on the radial coordinate. Appendix B More often than not, these rays with direction (8,4) will Numerical Computation of the Three-Dimensional cross the reacting space having no coincidence with points Attenuation. Figure 8 illustrates the case for a point like corresponding to the defined mesh for the axial and angular I. Let us define a new integration variable along the cylindrical coordinates ( q and w ) . Hence, for each point direction p , but measured from the point of incidence I. in the radial mesh (for example, in Figure 8, y = y,,) the In dimensionless form: concentrations must be averaged along the axial direction first (for example), over the distance between ql and q+1, T = [(P - &)/rRl 03.1) and subsequently over the angular distance between Wk Thus, at P = PI, TI = 0, and at p = P R , TR = [ ( A P R ) / ~ R ] and wk+l or, equivalently, between Bk and &+I. In the (Figure 91, which means that TR measures the dimenfigure, for instance, one must obtain first, two average sionless distance that has been traveled by the ray from values: one for B = P k , between q = q1 and q = ql+1, and, the point of incidence in the reactor volume to the point in a similar way, another for B = &+I. Subsequently the of entrance at the reactor wall. values for B k and P k + l must be averaged once more. In all To compute the species concentrations at each point of cases the average values are obtained by application of the ray trajectory, one can always chose points along the the lever rule. In equation form, and for the general case variable T (in the new spherical coordinate system with of yn:
-
Ind. Eng. Chem. Res., Vol. 32, No. 7,1993 1351
Y"
0
YI
(0)
Lt
II
I
I TI
yn
(b)
Yncl
!!
-
YR'1
N
Figure 10. Ray attenuation mechanism: (a) radial-angular mesh; (b) radial-axial mesh.
YJ'YI
0
Y"*
Y" Yl
(b)
Figure 11. Projection of the radiation attenuation process on the (r,6) plane for (a) I6 - 611< */2 and (b) I6 - 611 > d 2 .
I (yI,qI,PI) to one close to it, having a projection on the said plane that falls on one point of the radial grid; in Figure 10a this projection is represented by T'; it goes from I to G and the points on the radial grid are 71 and Y n , respectively. From this figure it follows that, for the general case with y = Y n ,
+ sin28 - 2T yr sin 0 COS[T - ($ - &)I Solving for T ,
y2 = 7:
$(qn,flk,Yn)(@k+l-
8,)
(B.4) AP The index n indicates that we are pivoting on the radial coordinate a t the point yn of the radial integration mesh. To facilitate the procedure and since we are using the radial direction as our pivot coordinate, it is convenient to project the distance Ton a plane perpendicular to the reactor tube axis. For a ray with direction (e,$) the said projection is
T = T sin 0
(B.5)
From Figure 10 it is clear that
qn = + T, cos e 03.7) In eq B.6 we use the positive sign when $ > PI and the negative one when $ < fir. With eqs B.1-B.7 we can compute the concentration of the absorbing species at selected points along the ray trajectory expressed in the p, e,$ coordinate system from information of the said concentrations computed in a grid corresponding to the cylindrical coordinate system (in general, $(qn,fln,yn) when the pivot point is chosen in the y direction). We can now proceed to the second step and develop the set of equations needed to compute the distance along the variable T . On the projection plane defined before, we would like to know the value of the projection of T from the point
(B.8)
[r2- y: sin2($ - &)11/2) (B.9) This equation must reproduce, as a particular case, eq 12 of paper 1written for the case of a constant absorption coefficient. In this case, writing the result in terms of p and computing the distance from the point I to p = pg we get 1 Apg pg - pI = -{-rI sin 6 COS($ - PI) f [r2- r: sin2($ - /31)11'2) (B.10) Equation B.9 must be carefully analyzed, particularly because of the double sign of the term inside the brackets. Two different cases are possible: Case I (Figure I l a ) : I$ - < n/2. It is clear that, for instance, for y = 71,TI = 0 and TZ< 0. Hence, only the positive root gives a valid result and the value of Tis given by [r2- y:
sin2($ - /31)11'z) (B.ll) The integral of eq 11 can now be solved. If we define
ATn = Tn+,- Tn
(B.13)
= (&In YI = ( W n I
(B.14) (B.15)
y
1352 Ind. Eng. Chem. Res., Vol. 32, No. 7, 1993
[n2- n: sin2(@-
(B.16)
[n2- n: sin2(@- /31)11'2] (B.17) Then
Case ZZ (Figure I l b ) : I@- /311> 7r/2. In this case, for y = 71,T1 = 0 and T Z> 0, and both values must be considered. However, we must pursue the analysis and look for the sign of t9 = [y2- y: sin2(@- ,411 (B.19) If 9 < 0 there not exists a real solution. This means that the projected ray (T') does not intercept any point on the radial integration mesh. This situation never happens if (6 - 01) = 0 because then t9 > 0. The limiting case occurs for 9 = 0. In this case, the projected ray (passing through the point I) is tangent to the circumference corresponding to a point in the radial integration grid (for example y = yn*). From eq B.19 we get y n . = y* = yI sin(@- BI)
(B.20) As can be seen in Figure l l b , the projected ray will not very often be tangent to this circumference and consequently, in general, an interpolation between two very close points of intersection will be required (points defining a secant). Then, the integration of eq 11 results:
n=nJ
n=nR
In this equation, nJ is the point of the first intersection of the projected ray with the circumference corresponding to y = 71. When computing the first term, one must notice that n decreases from nI to n* and consequently ATn = T,1 - Tn. In this case, the equation to be used for calculating T is
[n2- n: sin2(@- /31)11/2) (B.22) Figure l l b shows the reason for preceding the value of 29 with a negative sign; the same radial coordinate yn generates two roots; the first one corresponds to eq B.22. The second term of the right-hand side of eq B.21 should correspond to the value of y = yn*. This would be the special (but not very frequent) case when n = n*; then the solution for T is precisely the tangent to one grid point in the radial mesh and Tn*= = K*. When this situation does not occur, the position of T i n the radial coordinate is defined by two close points lying on the secant line. These two values for T are given by T;. = +n1 sin 6
cos(@-
[(n*)'- n: sin2(@- j31)31/2)(B.23) Between n* and nJ and between nJand nR one must use ATn = Tn+1- Tn and eq B.16, i.e., t9 with the positive sign.
Figure 12. Variables involved in the calculation of the concentration at the center of the reactor.
Appendix C Concentrationat the Reactor Center. The problem can be solved if one transforms the mass conservation equation in the vicinity of the center as follows (Ozisik, 1980):
This equation is then written in terms of finite differences, using equal increments (Ax = Ay) in the neighborhood of the center. After substitution, the concentration at the center of the reactor can be calculated from
+ Pe Ge(Ayl2[QAV+ 2(1- y2)$.1
--
(C.2) Pe G e ( A ~ ) ~ [ 2 ( 1y2)1 4A7 In eq C.2 is the dimensionless center line concentration corresponding to the previous interval in the axial direction, and $C is given, as indicated in Figure 12, by
rat,+
+
+
Literature Cited Agrawal, A. K.; Peckover, R. S. Nonuniform Grid Generation for Boundary-Layer Problems. Comput. Phys. Commun. 1980,19, 171-178. ALCOA. Aluminum Company of America. Technical Bulletin, 1964. 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., Ma.; Gordon and Breach Montreaux, Switzerland, 1986;Chapter 8,pp 339512. Cabrera, M. I.; Alfano, 0. M.; Cassano, A. E. Selectivity Studies in the Photochlorination of Methane I Reactor Model and Kinetic Studies in a Non-isothermal, PolychromaticEnvironment.Chem. Eng. Commun. 1991a,107,95122. Cabrera, M.I.; Alfano, 0. M.; Cassano, A. E. Selectivity Studies in the Photoreactor of Methane 11: Effect of the Radiation Source Output Power and Output Power Spectral Distribution. Chem. Eng. Commun. 1991b, 107,123-150. Cabrera, M. I.; Alfano, 0. M.; Caesano, A. E. Non Isothermal Photochlorinationof Methyl Chloridein the Liquid Phase. AIChE J. 1991~, 37, 1471-1484. Cerdl, J.; Marchetti, J. L.; Cassano, A. E. Radiation Efficiencies in Elliptical Photoreactors. Lot. Am. J.Heat Mass Transfer 1977, 1 , 33-63. Claril, M.A.; Irazoqui, H. A.; Cassano, A. E. A prior Design of a Photoreactor for the Chlorination of Ethane. AIChE J. 1988,34, 366-382. De Bernardez, E.; Claril, 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. Harris,P. R.; Dranoff,J. S. AStudy of Perfectly Mixed Photochemical Reactors. AIChE J. 1965,11,497. Hatchard, C. G.;Parker,C. A. ANew SensitiveChemicalActinometer. 11. Potassium Ferrioxalate as a Standard Chemical Actinometer. Proc. R. SOC.A 1956,518-536.
Ind. Eng. Chem. Res., Vol. 32, No.7, 1993 1353 Iwasaki Electric Co. LM. Technical Bulletin, 1975. Kalnay de Rivas, E. On the Use of Nonuniform Grids in FiniteDifference Equations. J. Comput. Phys. 1972,10, 202-210. Murov, S. L. Handbook of Photochemistry; Marcel Dekker: New York, 1973;pp 119-123. Ozisik, N. Heat Conduction; Wiley: New York, 1980; p 503. Parker,C. A. A New SeneitiveChemicalActinometer. I. SomeTriaIa with PoWsium Ferrioxalate. Proc. R. SOC.A 1953,104-116. Parker, C. A.; Hatchard, C. G. Photodecomposition of Complex Oxalates. Some Preliminary Experiments by Flash Photolysis. J. Phys. Chem. 1959,63,22. Peaceman, D.Fundamentals of Numerical Reservoir Simulation; Developmenta in Petroleum Science: Elsevier: New York, 1977; pp 120-127.
Romero, R. L.; Alfano, 0. M.; Marchetti, J. L.; Casaano, A. E. Modelling and Parametric Sensitivity of an Annular Photoreactor with Complex Kinetics. Chem. Eng. Sci. 1983,38,1593-1605. Tymoachuk, A. R.; Alfano, 0. M.; Cmano, A. E. The Multitubular Photoreactor. 1. Radiation Field for Constant Absorption Reactors. Znd. Eng. Chem. Res. 1993,preceding paper in this issue. Received for review September 17,1992 Revised manuscript received March 2, 1993 Accepted March 5, 1993
