A REACTlON SOURCE TERM MODEL "FOR PHOTOCHEMICAL REACTOR SCALE-UP Thesis for the Degree of Ph. D. MICHIGAN STATE UNNERSH‘Y PAUL R. HARRIS 19:67 uumgmqlgrwmnzlllIIHIMJWHHHI L 81319 This is to certify that the thesis entitled A REACTION SOURCE TERM MODEL FOR PHOTOCHEMICAL REACTOR SCALE-UP presented by Paul R. Harris has been accepted towards fulfillment of the requirements for Ph . D . degree infihgmigaLEngineering Major 'professor Date September 29L 1967 0-169 “'Vl“ ”I A REACTION SOURCE TERM MODEL FOR PHOTOCHEMICAL R EAC TOR SCALE -UP BY mr Paul RLllHarris AN ABSTRACT OF A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1967 ABSTRACT A REACTION SOURCE TERM MODEL FOR PHOTOCHEMICAL REACTOR SCALE ~UP By Paul R. Harris Recent investigations into photochemical reactor design have emphasized the determination of the reaction mechanism as necessary information for photoreactor scale-up. There are reasons to believe that such an approach may not be necessary. One reason is that empirical methods, using little knowledge of mechanism, have worked well in many other reactor design situations. Another is the fact that a photochemical mechanism study is often a long and tedious process which takes up valuable time during development of a process. In this work an empirical model for characterizing the reaction source term in photochemical reactors has been proposed and given theoretical basis. It was developed from consideration of three common causes of deviations from the Einstein photochemical equivalence law, namely reactions other than the reaction of interest, chain reactions, and dark reactions. This model does not require knowledge of the reaction mechanism. Procedures for use of this model for photoreactor scale -up in both monochromatic and polychromatic cases were developed. They are given in detail for the geometry used in the experimental part of the investigation. For industrial use in scale -up the model requires source intensity data, reaction medium absorption data and monochromatic, small-scale: reaction data. PAUL R. HARRIS The model was tested experimentally using a tungsten source which causes reaction within the wavelength region 300 to 600 milli- microns. The reaction system used was the decomposition of 0. 006M potassium ferrioxalate in 0.1N sulphuric acid, a nonmchain reaction, which has been used previously as an actinometer. A series of six reactors was used. They were of a unique cone shape with the light source located at the apex. This geometry gave a very strong geometric sensitivity to the local radiation intensity. The reactors ranged in volume from 19. 35 to 2654 cm3. Runs were carried out batchwise. It was found that conversions were of the order of 1. 5% or less, making the reaction seudo zeroeth order in concentration. It was found that the production rate increased by 23. 5% from the smallest to the largest reactor. Source intensity data and reaction medium absorption data were also taken to test the model. The experimental data were correlated in terms of four different quantum efficiency distribution models. The poorest correlation was obtained for the model which has been used in all previous polychromatic studies. This model assumed that the quantuzn efficiency was unity for all wavelengths. The next poorest correlation was obtained for a quantum efficiency distribution taken from all previous literature data on the potassium f‘errioxalate system. The best correlation was obtained from a step down function quantum efficiency model which assumed that the quantum efficiency was unity up to a .. certain critical wavelength at which point it dropped to zero. Almost as good a correlation was obtained from PAUL R. HARRIS an estimate of the quantum efficiency made fr om literature data at O. 006M ferrioxalate only and from absorption data on the reaction medium in the high wavelernth region. Plots were prepared of the dimensionless local reaction rate, the dimensionless average reaction rate, and the dimensionless production rate as functions of the dimensionless path length. It was found that the weakly varying dimensionless production rate gave a. much more severe graphical. test of the experimental data than the strongly varying dimensionless average reaction rate. It was concluded that variation of quantum efficiency with wavelength must be considered in order to confidently scale up photochemical reactors. This fact had been neglected in all previous investigations. It was also concluded. from ‘i.heor.:ti:'al considerations that the method developed for averaging the absorption coefficient over wavelength was the proper one. This method was given for spherical geometry. Previously, the average absorption coefficient was found by averaging over the incident intensity of the source. It was suggested as fut-are worn that a theoretical model of quantum efficiency dependence on wavelength he developed. The need for such a model was clearly shown from the data correlations. A REACTION SOURCE TERM MODEL FOR PHOTOC HEMICAL R EAC TOR SCALE -UP BY 11' 4 Hi" Paul RQL Harris A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOC TOR OF PHILOSOPHY Department of Chemical Engineering 1967 G L+' (a? ‘+& :9 3_§rc€ DEDICATION This thesis is dedicated to the memory of the four men who had the greatest influence on the author's character: Edbert Harris Joseph Benamy Barney Medintz Fred Lanoue ii ACKNOWLEDGEMENTS The help of the following peOple is acknowledged with thanks: Mr. W. B. Clippinger and Mr. D. L. Childs for constructing most of the experimental equipment; Mr. R. E. Rose and Mr. L. J. Keith for constructing the electrical system; Mr. J. F. Holland for providing facilities for measuring the intensity distribution of the experimental light source; and Dr. M. H. Chetrick, my research adviser, for his guidance and counsel during this entire investigation. This author also expresses his appreciation for the financial support of the Division of Engineering Research of Michigan State University. TABLE OF CONTENTS Page ABSTRACT DEDICATION. . . ............... . ..... . . . ii ACKNOWLEDGEMENTS. . . . . . . . ..... . . . . . . . . iii INTRODUCTION .................. . ...... 1 THEORY ..... ..... ..... .. 5 I. Experimental Method of Attack ........... 5 II. Reaction Source Term Model . . . . . . . . . . . . 9 A. The Monochromatic Case ..... . ..... . 12 B. The Polychromatic Case. . . - . . . . - - . . . 36 EXPERIMENTAL WORK. . . ....... . . . . ....... 48 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . 60 CONCLUSIONS ...... . ..... . . . . . ......... 84 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . 86 NOMENCLATURE . . . . . . . . . . . . . . . ...... . . . 105 BIBLIOGRAPHY ........ O O O O O O O O O O O O O O O O O 108 iv LIST OF TABLES Page 1. Dimensions of the Experimental Reactors , , , , , , , , 51 2. Relative Intensities of the Experimental Light Source . . 53 3. Experimentally Determined Quantum Efficiencies for the Decomposition of Potassium Ferrioxalate . . . . . . 57 4. Absorption Coefficients of 0. 006N Potassium Ferrioxalate in 5 millimicron Intervals From 300-600millimicron8.................. 58 5. Average Absorption Coefficient and Dimensionless Pathlength as a Function of Reactor Radius for the Experimental Reaction O O O O O O O I O O O O O O O O O O 61 6. Summary of ExperimentalReaction Data, , , , , , , , , 65 7. Correlation of Experimental Data for the Uniform Quantum Efficiency Model (Model 1) , , , , , , , , , , , 68 8. Correlation of Experimental Data for the Step Down Function Quantum Efficiency Model with a Critical Wavelength of 470 millimicrons (Model 2) . . . . . . . . 7o 9. Correlation of Experimental Data for a Quantum Efficiency Model Taken from all Available Literature Data (MOde1 3). O O O O O O O O O O C O O O O O O O O O O O 72 10. Correlation of Experimental Data for a Quantum Efficiency Model Taken from Available Literature Data and Absorption Data (Model 4) . . . . . . . . . . . 74 ll. Quantities Used in Calculating the Relative Intensity. Distribution of the Tungsten Source . . . . . . . . . . . 91 12. Results of Spectrophotometer Calibration Runs . . . . . 95 13. DataontheAReactor................... 97 14. DataontheBReactor................... 98 15. DataontheCReactor................... 99 160 Data on the D ReaCtoro o o o o o o o o o o o o o o o o o o 100 l7. l8. 19. 20. DataontheEReactor.......... ....... .. Data on the F Reactor. . . . ......... . ..... Numerical Quantum Efficiency Data for Model 3. . . . . Numerical Quantum Efficiency Data for Model 4, , , . , vi 103 104 10. ll. 12. 13. 14. 15. 16. LIST OF FIGURES Geometry of the cone reactor system with a point source I O O O O O O O O O O O O O O O O O O O O O O O O O O O Dimensionless local reaction rate vs. Dimensionless average reaction rate vs. pathlength with monochromatic light with 90 Dimensionless production rate v__s. dimensionless pathlength with monochromatic light . dimensionless pathlength with monochromatic light with p0 = 1 cm . . . . dimensionless =lcm.... Typical absorption coefficient XE: wavelength curve for a photochemical system . . . . . . . . . . . General shape of intensity XE: wavelength curve for a tungstensource.................. General shape of the product of absorption coefficient and intensity 173. wavelength . . . . . . . . . . . Quantum efficiency XE? wavelength for actinometric photochemical system. . Quantum efficiency vs. wavelength for industrial photochemical system . . . a typical a typical Typical curve for the average monochromatic reaction rate as a function of wavelength. . . . . . . . . . . . . . . Photograph of the experimental apparatus . . . Scale drawing of a cross-sectional view of reactor D . . . Electric circuit for the experimental light source . . . . . Average monochromatic reaction rates vs. wavelength in the largest and smallest experimental reactors assuming the uniform quantum efficiency model . . . . . . Summary of quantum efficiency models. . . . . . . . . . . Predicted curve for dimensionless local reaction rate vs. dimensionless pathlength using the Model 4 quantum efficiencydistribution. . . . . . . . . . . . . . . . . . . . vii Page 28 33 34 35 37 37 37 39 39 41 49 50 55 63 67 76 17. 18. 19. 20. Page Predicted curve for dimensionless average reaction rate \_r_s_. dimensionless pathlength using the Model 4 quantum efficiency distribution. . . . . . . . . . . . . . . 77 Predicted curve and correlated values for dimensionless production rate vs. dimensionless pathlength using the .Model 4 quantum—Efficiency distribution . . . . . . . . . . 78 Predicted curve and correlated values for dimensionless average reaction rate 172° dimensionless pathlength using the Model 4 quantum efficiency distribution over the experimental range of variables . . . . . . . . . . . . . . 79 Positioning of the light source within the reactor . . . . . 87 viii II . III. IV. LIST OF A PPENDIC ES Positioning of the Source Within the Reactor. . . . . . . . Source Calibration ..................... Calibration of the Spectrophotometer for Ferrous Iron . . Reaction Data ..................... . . . Quantum Efficiency Data for Models 3 and 4 , . . . . . . . ix Page 86 89 94 97 103 INTR ODUC TION Though photochemical reactions have been known and used for quite sometime as a laboratory tool of the physical chemist, their use in production of industrial chemicals has not been exploited until recently. They have been used in several commercial chlorinations, in the reaction of hydrogen sulfide with olefins to yield mercaptans, in the production of vitamin D in milk, in poly- merization initiations, and sterilizations (1). In the past few years, the Japanese have developed a photochemical technique for making cyclohexanone oxime (used in the manufacture of nylon) from cyclohexane and nitrosyl chloride (2). This one step process has replaced a two step thermal reaction process used previously. It has been pointed out that light sources could have an obvious advantage in terms of selectivity because light can be distributed over a narrow band of energies whereas thermal energy bands are very broad (Z). This point will be discussed further in the theory section. Though photochemical techniques have been used for industrial production, many photochemical reaction schemes have not reached commercial scale. Many (1-4) feel that this is due largely to the lack of adequate design procedures. The situation has been aggravated by the security measures of industrial firms working on photochemical processes. They refuse to make information available, especially on design procedures (1 -4). Design methods are slowly being developed by other interested parties such as private research organizations, government laboratories and academic institutions. The purpose of this investigation was to develop a model for the photochemical reaction source term which would be successful in designing photo-reactors. Basically there have been two approaches to logical photo- chemical reactor design. One which has gained great favor recently is a more fundamental approach, based on a knowledge of the reaction mechanism (4). The other, a more empirical approach, gives only little consideration to mechanisms (3). Proponents of the former method have been sharply critical of the latter method especially in the case of chain reactions. They argue that photo- chemical reactions are very mechanism dependent. A similar controversy arose over a decade ago in the case of heterogeneous catalytic reactor design. However, Weller (5) showed that simple empirical power law kinetic models could make predictions as accurate as those of far more sophisticated Langmuir-Hinshelwood models based upon mechanism. The fact that most catalytic reactors designed today are operating without a thorough knowledge of the mechanism, is a great attestment to the value of this sort of empiricism. Another point is that chemical business economics are being based more and more upon new products, products which are valuable because they are "new”, i. e. , the fact that their technology is not well developed. Time then is becoming a more important factor in the development of a new product. Large profits are made early before the product's technology is well understood. Empiricism is then justified by its economic value; it saves time during develop- ment giving a greater return on investment. Had industry waited to learn of the mechanism, most catalytic processes would still not be in operation. Therefore, the empirical method developed in this study is justified. First, it works like Weller's empirical method for catalytic reactor design and secondly, it will save considerable amounts of develOpment time which should give a substantial increase in return on investment. This study has grown out of another investigation of photo- chemical reactor design (3). The latter suggested several improvements, both experimental and theoretical which could help in the search for a rational design procedure. First, the earlier experiments used a cylindrical annulus with a linear source at the center, aligned with the cylinder axis. In order to make the problem one—dimensional and thus experimentally much more tractable, it had to be assumed that all radiation from the source was purely radial. This assumption is of course unrealistic, since radiation is skewed from a linear source. Jacob and Dranoff (6) tried to remedy this by putting spaced mica disks around the source. It was decided in this investigation to use a point source in a spherical geometry. This eliminates the problem entirely since no line can be skewed to a point. Another experimental advantage of a spherical geometry is its geometric sensitivity, namely the stronger increase of volume and the stronger decrease of radiation intensity with reactor radius changes. Another experimental problem was in the reaction system used. The first study treated a chloroplatinic acid hydrolysis and measured conversion by the electrical conductivity of the hydrochloric acid produced. Electrical conductivity measurements, though convenient, are not highly accurate being especially sensitive to impurity and temperature errors. For this reason it was decided to use one of the most accurately measurable photo- chemical systems known, the decomposition of potassium ferrioxalate (7). Other reasons for selecting this system will be discussed in the Theory section. Several theoretical improvements were suggested also. The analysis has now been based upon a continuous wavelength distribution. This has required a new, more general expression for the reaction rate source term. A proper method for averaging the absorption coefficient of a medium over wavelength has finally been formulated. In this investigation, special attention has been given to finding out how much knowledge of the variables associated with the light is required to accurately design a photochemical reactor. Of particular interest was wavelength dependence of the absorption coefficient, light intensity, and quantum efficiency. The quantum efficiency dependence was of special interest since it was neglected entirely in all previous investigations (3). 5 THEORY 1. Method of Attack The interesting and complicated phase of photochemical reactor design, which distinguishes it from conventional reactor design, is the effect of the variables associated with the light on the reaction rate expression. This is a source term (actually a sink) in the material balance. * If one were interested in studying the effects of these variables, he would choose an experimental system in which other transport phenomena associated with conventional reactor design had no effect or as little effect as is experimentally possible. For this reason a single phase, isothermal, perfectly mixed reactor was chosen for this study. Thus the momentum and heat balances have no effect on the study. This system further simplifies the material balance since there are no diffusion terms and no concentration gradients, making the material balance simply a first order ordinary differential equation in time rather than a partial differential equation which is second order in position and first order in time. At this pomt the experimental situation is simplified to the following differential equation. __ I Fci -FC ~RV : “gr-)- (1) >{‘Actually one should say here "material balances". There should be a balance on each component. But as is often done in conventional reactor design it can be assumed that the source term is a function only of the conversion, i. e. , that the material balance can be defined by how much of the reactant has been converted. The extension to where F is the flow rate Ci is the input concentration of reactant V is the reactor volume t is the time C is the output concentration of reactant R is the overall reaction rate in the reactor In this equation R is some unknown explicit function of C and of the variables associated with the light. In principle F, Ci and V could all be independently variable functions of time. But since we are interested in the dependence of R on the light variables and not on its changes with F, C1 and V we will further simplify the experimental system by holding F, Ci and V constant with time (or as constant as is experimentally possible). Our differential equation is now reduced to r _ £19.. FCi~FC-RV -V dt (2) where F, Ci’ and V are known constant parameters in any experiment. From this point on the magnitude of R dictates the experimental method. If R is very large, one would operate in reacting systems in which the reaction rate expression is strongly dependent on more than one concentration would require a material balance on each component. Since the effects of the variables associated with the light are of primary interest, in the discussion that follows only those reaction rate expressions which are a function of only a single component concentration (and thus conversion) will be treated. Obvious multicomponent extensions can be made. a steady state flow manner with a high F . dC/dt would then be zero and F would be given by: - F R : V (Ci .. Css) (3) If R is in an intermediate range, one could operate in a semi- batch fashion and test the differential equation (2) in its most general form. This would involve studying the approach to steady state. If F is very low, as it was in this investigation, one can operate in a batch fashion with no flow at all. This reduces the differential equation to .___ dC R—"dt (4) From batch experiments the differential equation can be integrated to give E as a function of C . The interest in this investigation was more in F as a function of the variables associated with the light, especially in those variables which changed the production rate, P, (equal to ~FV) in the scale up of different-sized reactors of the same shape. In order to make a comparison of the effects of the light alone on P one would be required to measure §{C) (and thus P{C}} experimentally (changing the constant V for each rea‘ctor) and compare the P's obtained in the various sized reactors at constant values of C . Since If as a function of C was not of great interest in this investigation, and since the reaction was quite slow, only the initial reaction rate was measured. Conversions were very low for all runs (of the order of l. 5%) so that C and thus F did not change much during the course of the reaction. Thus II was pseudo-zeroeth order in concentration and could be approximated by i = - —-— (5) This simple formula was used to correlate the experimental data. In this investigation a mathematical model was developed for P (or IT) as a function of the variables associated with the light, and this model was tested (with respect to scale-up of a photochemical reactor) by the experimental method just described. It was tested on a particular set of reactors, namely those of a cone shape with a point source at the apex of the cone. This set was chosen because it provides the most accurately defined experimental conditions yet devised to test photochemical models of all kinds. 9 II. Reaction Source Term Model It is helpful to consider a photochemically reacting system as a thermodynamic system composed of two weakly interacting thermodynamic subsystems. These two subsystems are the reaction medium and the light or photons. If only a chemical system were being considered and local equilibrium were assumed, then the thermodynamic state at any point could be specified by giving the temperature, the pressure, and the concentration of all but one of the components in the reaction medium. This is true for a. simple, single phase thermodynamic system of n components since any of its intensive properties (including reaction rate) is a function determined by any n+1 other intenSive properties (8). As stated above one must assume local equilibrium or a "quasi-steady thermo- dynamic state" in order to treat a chemical system as a thermo- dynamic system. This assumption means that the distribution of energies of the molecules of the system at any point is essentially a thermostatic distribution so that the system at any point can be defined by the thermostatic intensive prOperties there. Thus the system is unaffected by gradients. Another assumption usually made in treating chemical reacting systems (though not necessary for thermodynamic treatment) is the so called "stationary state hypothesis" of reaction intermediates. Roughly it says that activated species are very short-lived so that they do not build up to large concen- trations which will affect the stoichiometric dependence of the consumption of reactants and the formation of products. In 10 mechanism postulations, this assumption usually appears as a statement that the time derivative of each intermediate is negligibly small in comparison to reactant and product derivatives. It allows one to say that the concentration of the intermediate does not vary much during the course of the reaction and thus its concentration can be substituted for in terms of concentrations of reactants and products whose concentrations vary during the course of the reaction. Thus, the reaction rate expression can be written in terms of reactants and products and not in terms of intermediates. Note that the two assumptions (the quasi-steady thermodynamic state and the stationary state hypothesis) are not entirely independent since the active intermediates are highly energetic chemical species so that if their concentrations were high they might significantly perturb the thermostatic distribution of energies. The thermodynamic state of the light alone is, by the analogous argument for chemical thermodynamic systems, determined by two thermodynamic intensive variables. These will be chosen for the moment as the intensity and the energy of the photons. Note that light does not necessarily have a themiostatic distribution of energies. Therefore, photons of different energy must be considered as separate thermodynamic systems. Photons of one particular energy (light of a single wavelength or so-called monochromatic light) will be considered first. Then intensity and energy (or wavelength) determine the thermodynamic state of the system. The photochemically reacting system is considered as a "slight perturbation” of the chemically reacting system. This is so 11 slight that one can still make both the quasi-steady state thermo- dynamic assumption and the steady state hypothesis. The two additional thermodynamic variables required are the intensity of the light, or more conveniently as will be shown later, the product of the intensity and absorption coefficient of the medium, and the energy, or more conveniently the wavelength. Note also that the absorption coefficient is not an additional intensive variable since it is already determined by the wavelength of the light, the concentrations of the medium, and the temperature of the medium. A photochemical reaction can be pictured as a reaction between photons and unactivated molecules to form activated molecules which then react in some way to form products. The quasi-steady thermodynamic state assumption means that in this process the intensity (or energy) will not become so high that the thermostatic distribution of energies will be appreciable affected. From the steady state hypothesis, it is known that in this process the intensity will not become so high that the concentration of intermediates will become appreciable. However the intensity can vary in space due to absorption (and other‘processes) and the concentration of intermediates will vary in space even if the reactor is perfectly mixed with respect to reactants and products. If the reactor were perfectly mixed with respect to reactants, products and photoactivated intermediates then there is no effect of intensity variation at all and the reactor cannot be distinguished from a conventional perfectly mixed chemical reactor once the total 12 absorbed intensity is known. In effect, making the concentration of activated intermediates uniform throughout would make intensity uniform throughout the reaction medium. This point will be returned to later in the discussion of the monochromatic case. A. The Monochromatic Case It may be assumed that the source of light is of fixed intensity and wavelength. The polychromatic case will be treated as an extension of the monochromatic case. Assume that the reactor is perfectly mixed with respect to concentrations of reactants and products but unmixed with respect to intermediates. It is assumed also that the reaction has a uniform temperature and that this temperature is fixed. From the above discussion it is known that the rate of reaction at any point in the reaction medium is a function only of the intensity at that point and the reactant and product concentrations (or reactant concentration and conversion). The intensity is a vector type quantity and as such camot couple directly with the scalar quantity concentration by the Curie Theorem (9). However the magnitude of the intensity can couple with the concentration in the reaction rate expression since it is a scalar. From the theories of irreversible thermodynamics it can also be shown that a photochemically reacting system is inherently a non-equilibrium system. Since the magnitude of the intensity *Light flux in a single direction is a vector in that it has magnitude and direction. But combinations of light fluxes from several sources do not add as vectors to form a new "intensity". Only the magnitude of the so-called intensity is defined, and it is defined as the algebraic sum (all positive terms) of the magnitudes of the single intensity vectors through that point. 13 is now a thermodynamic force (like the concentration or chemical potential) and Since it varies throughout the reaction medium due to absorption (and other processes) there is an inherent (an in real cases large and nonlinear) gradient within the system making thermostatic equilibrium impossible. Thus a photochemical system is one with a reaction rate gradient even though it may be perfectly mixed with no concentration or temperature gradients. Perfect mixing does not include concentration gradients of active inter- mediates which are not included in the thermodynamic specification of the system. Consider now the simplest of reactions, namely a single reaction of stoichiometry aA -' bB + cC + . . . . . . . . . . (referred to later as the "reaction of interest”) which obeys the Einstein photochemical equivalence law exactly, i. e. , each mole (or einstein) of photons absorbed causes one mole of reactants to form a stoichiometric number (b/a, c/a, etc.) of moles of each product. For this system the rate of reaction, R , at any point in the system would just be the rate of light absorption which is given by .3 ll R=Hl (6) where R is the reaction rate in moles of A decomposed per unit volume per unit time or a/b times the moles of B formed per unit volume per unit time, p. is the absorption coefficient of the medium in (unit length)—l, and [TI is the magnitude of the intensity in moles of quanta per unit area per unit time. 14 Assuming a quasi—steady thermodynamic state and assuming the stationary state hypothesis is valid, deviations from the above law can be attributed to three causes: 1) absorption of light which does not cause the reaction of interest, 2) chain reactions, and 3) dark (thermal) reactions. Light absorption due to processes other than the reaction of interest will be treated first. A light photon, when it is absorbed, increases the energy level of the molecule which absorbs it (by an amount equal to hc/k ). The activated molecule can then deactivate by two processes?-E< l) by degradation of the kinetic energy of the molecule by collisions with other Inolecules (thus increasing the thermal energy or temperature of the medium. The medium must be correspondingly cooled to hold the temperatureconstant.) 2) by reaction through decomposition of the molecule or through collisions with other molecules. Deactivation by the former process will always make R less than the value given by equation (6). Note that any miolecule of product, solvent or even reactant can absorb and deactivate by this thermal process. In carrying” out industrial reactions one would of course try to minimize absorption with thermal deactivation since it wastes light. Deactivation can also occur by reemitting another photon at the same wavelength (resonance radiation) or'at a lower wavelength after partial deactivation by another process (fluorescence). -.Neither is a common phenomenon in photochemical reactions and thus will not be considered further. y—a ()1 Assume that absorption and deactivation by thermal means is negligible. Then all deactivation would occur by some form of reaction. Now if all absorption is by reactant molecules of interest and all deactivation is by the stoichiometric reaction of interest, the equation (6) will give the reaction rate. Deviation from the above statement can occur in two ways. First of all deactivation can occur by reactions other than the reaction of interest. This would mean a reaction in parallel with the reaction of interest. This would still give R exactly in terms of the amount of reactant consumed by equation (6) but would give an R lower than that given by equation {6) in terms of the amount of product formed according to the reaction of interest. Secondly, deviations can occur by further consecutive photochemical reactions of the products of the reaction of interest. This would decrease R below that given by equation (6) in two ways, by using photons in a reaction other than the reaction of interest and by decreasing the amount of product formed. Because of reactions other than the reaction of interest, the definition of R given by equation {6) is ambiguous in that R is not the same for the consumption of reactant as it is for the formation of product. Rc will be defined as the rate of consumption of reactant in moles of A consumed per unit volume per unit time and Rf will be defined as the rate of formation of product in a/b times moles of B formed per unit volume per unit time. The absorption coefficient of the medium can be divided into thermal deactivation and reaction deactivation components by the following equation 16 H = P~ +Ht (7) where p r is the absorption coefficient for absorption causing chemical reaction when deactivation occurs in (unit length)‘1 “t is the absorption coefficient for absorption which is dissipated as thermal energy in (unit length)’1 The reaction component u r can be further divided into a component of the reaction of interest, a component of the reactions parallel to the reaction of interest and a component of reactions in series with (or consecutive to) the reaction of interest by the following equation: p'r : IJ'ri‘lpp'rp+ld'rs (8) Combining equations (7) and (8) yields p. : H"ri+}‘1‘r°p+P'rs‘ihp't (9) The rate of consumption of reactant would then be the sum of the rate of consumption by the reaction of interest plus the rate of consumption by reactions in parallel with it. Rc=uril1l+p~rplll (10) or in terms of u and a coefficient P~ -+H _, RC=M f1 +IP+ H1! (11) I~J'ri ILLrpp'rs H't The rate of formation of product (in terms of b/a moles) is Rf=u.|Il-p III (12) 17 or in terms of u and a coefficient H -'H _\ r1 rs R = u { } III (13) f Hri+Hrp+Hrs+Ht Thus equations (11) and (13) can be rewritten in terms of a quantum efficiency or quantum yield, § RC = H If! éc (14) .J Rf = bill, if (15) where g3 _ p'ri + lJ'rp C H'riJerp+p'rs+p't and é : I“LEVI u Hrs f lJ'ri+Hrp+l‘Lrs+Ht Obviously Rc and Rf are not the same and differ by the ratio 59 : Hri + Hrp Rf H'ri - Hrs When “rs and Mt are negligible, Rc = R, when Hrp and ”t are negligible H --H R :(r1 r5 f H'ri+Hrs )R, and when Hrp and [4. rs are negligible then 18 It should be noted also that the quantum yield as defined here is a quantity which is independent of intensity since the p. 's are all independent of intensity. It should be remembered also in this connection that a quasi—steady thermodynamic state has been assumed. If the intensity were so high that this assumption could not be made then the deve10pment just given would not be valid and the quantum efficiency would depend upon intensity. However under ordinary circumstances this assumption would be quite valid. Comparable conditions in thermal chemical reactions occur in the case of very fast, high temperature reactions where quasi-steady state cannot be assumed because there is not a thermostatic distribution of energies. In very high intensities the concentration of activated molecules will build up by primary acts faster than it can decrease by the various deactivation processes. This will of course cause an appreciable transient concentration of activated species. What has been assumed by the stationary state assumption is that no such appreciable transient build up of activated species will occur which will affect the stoichiometry of the reaction of interest, parallel reactions, or series reactions. Another way to state both the stationary state hypothesis and the unmixed intermediate assumption would be to say that all deactivation processes occur very quickly in comparison to rates of absorption. Since deactivation of an excited species requires essentially no activation‘energy and steric factors for reactions of electronically excited molecules should be the order of unity. The deactivation rate should be of the order of the collision rate of activated molecules with other 1‘? molecules. The concentration of activated molecules at any point in the system should build up fast to a small value since the rate of light absorption is small in comparison to the rate of collision between activated and unactivated molecules except at very high intensities. So far, in considering effects due to absorption which does not cause the reaction of interest, it has been implied that reactions are caused through a deactivation process. This is not necessarily the case. Reactions can occur in which an activated molecule of reactant collides with another molecule forming a molecule of product and a different activated molecule. This new activated molecule collides with another reactant molecule to form the original activated molecule and another molecule of product. The above statements refer to the simplest form of a chain mechanism which is the second cause for deviations from equation (6). In the case of photochemical chain reactions the usual approach has been to formulate a mechanism based upon a photo- chemical initiation step given by equation (6}, a series of chain propagating steps of ordinary chemical reactions involving activated species, and a termination step involving a recombination of two activated species (with or without a third body) to form deactivated species plus some thermally excited species (species with increased kinetic energy). * All deviations due to processes previously mentioned are neglected here. 3k The termination reaction can of course be heterogeneous. This would make R dependent on surface to volume ratio. But as long as perfect mixing, the quasi-steady thermodynamic state, and the 20 The equations are then solved for R uSing the stationary- f state hypothesis for all activated chemical species. This leads to an expression for R which is of the form _1 RzR =Rf:(ulll)mf(CA,C c C B, C, > (16) inerts where m is a small rational fraction (proper or improper) and f is some function of the concentration of all the various non- activated species in the system. Assuming that the reaction of interest is a chain reaction and that none of the processes of deactivation previously mentioned are important (this means of course that thermal termination gives a small Ht)’ one can rewrite equation (16) in the form of equations (14) and (15) - l’fl u 9 (17) 50 H chain where i =(M‘fhm‘1uc c c chain A’ B’ C’ ) inert Note the é . has a weak (usually m is less than 2 so that m-l chain is less than 1) but finite functional dependence on I I I p . With chain reactions the assumption about unmixed activated species is not as good as before since chain molecules are less readily >l< _ (con't) stationary state hypothesis can be assumed, one could use some sort of pseudo-homogeneous expression with quantum yield (defined shortly) which is a function of surface to volume ratio. 2.1 U.) deactivated but it is certainly no les- ‘rvalfdi‘ than the stationary state hypothesis for a2-.. '“e epe :ies used in deriving equations of the form of equation (16). Usually échain is greater than one, otherwise the reaction would not are a chain reaction. We can .- l a define a chain reaction as any reaction for which R > |I| p. in the absence of parallel dark reaction in which the quantum efficiency <1) can be grea.,er than one. Now if the other deactivation processes occurred. thy nc~n==chain means) then RC and Rf would be given E‘s-y ...3 R: = II! H é: échagn (18) Rf = III u if «am <19) Note that the re ac isn rate can still be written in the form —_8 R : |i§ p t (20) d: r One can think of the pr sblem in terms cf three rates, the rate of activation by phi—s an -, the rate of deactim n by thermal degradation and reiU-cticn tc ti: :1 its, and the ..a'.e of mixing of activated species. Thes ta :j ' ' ticn makes the rate'of activation sacral to the rat . /: ion. The rate of activation is fixed by (actually prcpcrt'isnal to the intensity of the light. It The - ate of de a::t‘“t;°a":.i:;zr. is fixed by the concentration of the intermediate. The stati. . state assumption makes this rate fast so that the cones rtr‘aticn will be low even if the intensity is high. It is implied here alas that he Ia e of deactivation is so fast by this assumpt' :n that it s fas tar th 3:1 the. rate of mixing. Hill and Felder (10) have 3.21 di ed his pro elem in detail and it is concluded from their arguemerts that thein te: mediates are essentially unmixed, except to «ial (and easily modeled) cas e cf chain re-a- “ vans z** m ,4 l ard very long chain lengths, dii ‘ lifetime of reaction intermediates is es short 1.; times If the i . , would actually te ccame simpl- uniform over the entire :eacLsr. + Til D ‘h 1) 3 ("f .+ T. w t + k" : U“ 'T' S) a: ' 9 1" HQ . h .drodynamic mixing 1 .re complete y mixed the problem t1“ ea tion rate would be rig" ti :9- l1 (. (D 0.. F4' "J .r ‘ u'k * m S o 22 where i can be expressed as a product of a small exponential power of IT) p. (the exponent is zero for non—chain reactions and usually less than one for chain reactions) and a concentration factor. If there is more than one photochemical chain reaction such as photo- chemical chain reactions in parallel to the reaction of interest by more than one mechanism, then equation (2) would not be strictly correct, but would probably be adequate for design purposes (as will be discussed shortly). Similarly if several chain and non-chain reactions occurred one might still write equation (20). So far very little has been said about the concentration dependence of the reaction rate expression. Although it is not of primary interest in this investigation it is usually important in design so it will be considered. From a thermodynamic point of View, the reaction rate expression is a function of all the species present (excluding reaction intermediates). It would in general be a complicated function of them. However as with thermal chemical reactions with the usual boundary conditions of no product initially and all reactant and inerts initially, the reaction can usually be . written as a monotonically increasing function of reactant concen- tration or a monotonically decreasing function of conversion. This would not be true for example if the reaction of interest was a consecutive process in which the products of interest were interme- diates. The concept of order of a chemical reaction should be just as applicable to the local rate expression in photochemical reactors with intensity gradients as it is in conventional chemical reactors with concentration and temperature gradients. With a complex reaction, 23 even though it may have no maxima it could still have many inflection points. However, except under special circumstances in which the relative importance of various rates are of the same order of magnitude, these inflections will be so weak as to be unnoticed. Thus the concept of order seems to fit quite well here. It is suggested then that quantum efficiency be written as i = K(|I|M c: (21) where m and n are independent of temperature. Note that in most cases (without highly absorbing products or solvent) p. is approximately proportional to C by Beer's A law. Thus the true order of the reaction with respect to concentration would be (m + n). Since one would not expect the overall order with respect to concentration to be greater than 3 or less than 0, one would not expect n to be greater than (3 - m) or less than (-m). Since no is usually between 2 and 0, this means that n is usually less than 2 or greater than - 2. One would expect it to be less than - 1 only in the case of chain reactions. Similar arguments to those used for concentration dependence of reaction rate can be used for (1):.)rn-1 dependence of quantum efficiency even in the presence of multiple chain and non-chain reactions. A single non-linear parameter seems adequate for defining the shape of both the R versus (1p) curve at constant * CA and the R versus CA curve at constant (In) . The parameters are m and m +n, respectively. A constant is required for determination of the absolute magnitude. 2.4 The last reason for deviations from equation (6) is dark reactions. In general all dark reactions are minimized in commercial situations by operating at relatively low temperatures so that the reactions which interfere with the reaction of interest will be negligible. The reasons for utilizing photochemical techniques is that interfering thermal dark reactions decrease the selectivity for the reaction of interest. If dark reactions occur quickly in the series of intermediates between reactants and products they need not be considered (all deactivation steps are actually in this category). If a slow dark reaction occurs in the series of interme- diates before products of interest, the reaction is essentially a chemical reaction and can be treated as such. If a dark reaction occurs on the products of interest one gets the true rate of formation by subtracting the dark reaction rate from the rate found from equation (20). Similarly one subtracts or adds the rate of a dark reaction in parallel to the reaction of interest to the rate found from equation (20) depending upon whether it is a competing side reaction or the dark form of the reaction of interest. In any case measurements of dark reaction must be made if it is appreciable by carrying out the reaction under the same conditions in the absence of light. In the experimental part of this investigation dark reactions were known to be negligible. The following point has been reached for the monochromatic case. The source term R at a point is given by an equation of the form 25 R = u I‘I‘l i (20) where t = K(l‘flu>m"1 c: (21) where K m, and n are experimentally determined parameters depending upon the measured values of R, p, (Tl , and CA (or conversion with known initial concentration). One is interested in R as source term in three ways. It is desirable to know R itself at various points within the reactor, to know R, the average reaction rate over the entire reactor, and to know P the average production rate of reactor in molefloduct formed unit time , the three quantities are related by the following equations . R = (22) 11: dV RdV L— : -\!,-5 R dV (23) S dV V V. ml n P=SdP=5RdV—RV (24) V V or in terms of equation (20) R = p. I?! § (25) fi=$—Sul‘f|§dv (26) V ng ulTlédV (27) V 26 It was of particular interest in this investigation to see how the three quantities would scale up as a reactor was made larger in the direction of the light path if the model for R given by equations (20) and (21) was used. If a perfectly stirred reactor is used, H. (and CA) become independent of position and only IT) and § can vary with position. If further a non-chain reaction is used, the value of m in equation (21) would be 1 and the quantum efficiency, é , would be independent of position. In this investi- gation a non-chain reaction was used so that this was true. The model, therefore, was not tested for variations of § with IT) . Thus, equation (25) remains the same with § now not varying with position and equations (26) and (27) reduce to R" = fix?) I?) dV (28) V qu§5 l‘fldv (29) v For monochromatic light one must evaluate the integral in equation (28) and (29) to get R and P. To do this, the intensity with position must be known. This field is determined by Lambert's law which is written below in vector form. ..3 A, diV|1l=-H'Il (30) In this investigation the simplest, and most clearly approxi- mated in practice, geometry was chosen for the shape of the source. Namely a uniformly irradiating point source* at the apex of a cone 3:: This means that light comes out at the same intensity in all directions from the source. 27 which is a portion of a sphere. A drawing of the geometry is given in Figure 1. In this geometry with the spherical coordinate system shown the intensity components in the 6 and (1) directions are zero throughout the system. Therefore, Lambert's Law reduces to an ordinary differential equation, involving the radial component of the intensity (a scalar equal to the magnitude of the total intensity). 1d 2 ———- I =- 1 31 pde((3 p) u p ( ) * Equation (31) can be integrated from a source radius pO to outer radius p , as follows. I 2 -" p d

' ‘Note that though p0 can be infinitesimally small, the product pilp must still be finite. M(Note that light intensity is not only decreased due to absorption as shown in the exponential term, but also by the so called "inverse square law" decrease due to increasing illumination area as the radius increases. 28 POINT SOURCE \% P. Figure 1. Geometry of the cone reactor system with a point source. Z9 dV 2 p2 sin 43 do dd) d9 (35) Thus the volume of the reactor is ,ZTr ¢ 9 ' 2 . 2 3 3 V = p sm¢ dp d¢ d9 = §w(1-cos¢)(p -po) (36) o 0‘ 0 Substituting equations (33) and (36) into (28) an expression for R is obtained. .211' (I) .p p2 H§3 S 5 IP —§ expl-Mpwofl pzsin¢dp d¢ d9 —_ o o 0 p R _ 2 3 3 (37) §W(1-COS¢)(P '90) _ @190 pianmécos ¢){1- exp[-u(p-po)]} R z 2 3 3 (38) §W(1—COS¢)(P ‘90) 31 p2 ._ Po 0 R=§ -—3——-§ {l-eXpl-Mp-pofl} (39) p -po In a similar manner one can substitute equations (33) and (36) into (29) to get P. P = t Ipo pi (2w) (l-cos «M {1 - exp[ - p (p - po)]} (40) In order to treat the experimental data, these equations should be put ina certain dimensionless form. Thus >kIt should be noted here that the expression can be integrated analytically in the case of chain reactions with 4) within the integral sign, if m = n or m = n + 1/2 where n is a positive integer. All these solutions are infinite series except that for m = 1/2. This important particular case of a completely spherical reactor has been treated recently by Hormer and Wilkinson (11) for monochromatic radiation. 30 reference values for the reaction rate and production rate should be taken. The most convenient reference value for the reaction rate would be the maximum reaction rate, which occurs at p0 where the light enters the reactor. (This is not necessarily so in the very rare case of a chain reaction where m would be negative.) It is given by equation (34) setting p 2 p0 . Rmx = u 3 IPO (41) One can now define the dimensionless local reaction rate, R*, at the radius p by 2 R32 - R - :9 [ ( )l (43) ' - R " 2 exp - H‘ p '- p0 max p One can also define the dimensionless average reaction rate, R*, for a reactor of radius p by 13* : ii 3 Po .. R = 3 3“ {1- exp[ - (up-pen} (43) In the case of production rate the most convenient reference value would be the maximum production rate which would occur when all the light was absorbed within the reactor. Such a reactor would have to be infinite in radius, p . By lettering p --> 00 in equation (40) P is obtained. max P = é I p: (2n)- (1 - cos ct) (44) The dimensionless production rate for a reactor of radius p can now be defined by 31 *U P" = p =1 -exp [w (p-p0)] (45) max Note that for monochromatic light with a non-chain reaction that all three of the dimensionless quantities just defined are independent of both the quantum efficiency é and the incident intensity I . In chain reactions with monochromatic light these dimensionless quZntities would still be independent of Ip but not independent of § . It is desired in practice to correlafe the data in terms of some useful measure of the reactor size. The path length of the light, p - p 0’ seems the most useful measure of.reactor size. A form of pathlength which represents all sizes of reactors on a finite coordinate seems desirable. The simplest such form would P ' P o . . . . be (p _ 9037+ B , where B is some p031t1ve number With the dimensions of length. This dimensionless number would increase from O to l for (p - po)’s increasing from O to 00 , regardless of the positive value of B chosen. However it would be desirable for B to be related in some way to the amount of light absorbed, since this is strongly related to the other three dimensionless quantities. The absorption coefficient )1. has dimensions of reciprocal length, so the choice would be l/p. , the distance over which the intensity is decreased by a factor of e . Thus the dimensionless pathlength [>3 will be defined as 32 12* : (p --- pol 4: (p - 90) (46) T-l" Plots of R* and R’r" versus 1* are given in Figures 2 and 3 for p. values of 1/10, 1, and 10 cm"1 and p0 ofl cm. Obviously R* decreases more rapidly with 1* than does R* . The inflection points in the curves are due to the choice of dimensionless variable 3* . Had a dimensionless variable such as p/pO been chosen, there would have been no inflections but an infinite abscissa would have been required to show all sizes of reactors. In Figure 4 is shown a plot of Pi< versus 1* which, unlike R* and R"!< versus 3*, is a unique function independent of p. and p0 (for monochromatic light). This is because both P* and 1* are both dependent only upon '7' = u. (p — p0) , the so-called optical pathlength. Note that the inflection point in the P* versus 1* curve occurs at 3*- : . 5, the point where 7 =1 (or p .. p0 = 3:). Before going on to the polychromatic case, a few more comments might be made about the rate expression given by equations (20) and (21). One might think it more natural to write the equations in terms of the reaction of interest alone. That is Rzp.|‘f{§, (47) 1‘1 1‘1 _ . m-l n i - — Kridp'ri) CA (48) II The reason for not doing this is that p and § in terms of p. are much more easily measured experimentally in monochromatic 33 I. , , 0.6 dimensionless local reaction rate a): R O. O. O. 0.0 0.2 0.4 0.6 1.0 dimensionless pathlength, 1* Figure 2. Dimensionless local reaction rate 33' dimensionless pathlength with monochromatic light with Po = 1 cm. 34 0.8 0.6 dimensionless average reaction rate r8“ 0 0.2 -1 H=.lcm 0.2 0.4- 0.6 [.0 dimensionles s pathlength, l * Figure 3. Dimensionless average reaction rate 13.: dimensionless pathlength with monochromatic light with po = 1 cm. 35 I. 0.8 0.6 dimensionles 3 production rate P* o 0.2 O O 0.2 dimensionless pathlength, 1* Figure 4. Dimensionless production rate 2. dimensionless pathlength with monochromatic light. 36 experiments (both IJ. and § vary with wavelength) than IL ri and éri in terms of Hri . In. regard to experimental measurement of the R function it should be said that the experiments should be batch experiments with low conversion and very short cells over narrow wavelength ranges. This will keep the important quantities p. , é , from varying during the experiments due to variations in A I I I , CA and wavelength. B. The Polychromatic Case So far very little has been said about the effects of wavelength or energy of the quanta being absorbed. These effects are very important in the design of the reactor, for instance, in the choice of a source of light. Just as one can think of (ITI k HR) in a photochemical reaction as somewhat analogous to concentration in conventional chemical reaction, one can think of the wavelength (or the reciprocally related energy) as analogous to temperature in the conventional chemical reaction. In the region of energies of photons which generally activate an electron level in a molecule, the absorption coefficient pk generally decreases with wavelength as shown in Figure 5. Assume that the source has an intensity which is continuously increasing with wavelength. This is true for the tungsten source used in this investigation. It is illustrated in Figure 6. The product pix ITI .\ is thus given by a curve of the general shape of Figure 7. Thus it is known from equation (20) and Figure 7 that the reaction of interest will not be carried out below a certain wave- 37 absorption Figure 5. Typical absorption coefficient vs. wave- coefficient, _ “X length curve for a photochemical system. wavelength, A intensity 'I‘l Figure 6. General shape of I X intensity 2/3. wave- length curve for a tungsten source. wavelength, X i‘ P‘XI IX Figure 7. General shape of the product of absorption coefficient and intensity 2. wavelength. wavelength, X 38 length limit because no light of these wavelengths will be put into the system and will not be carried out above a certain wavelength limit because the light does not have enough energy to activate the molecules. The quantum efficiency for the reaction of interest varies in a similar fashion as shown in Figure 8. The flat region B of relatively high quantum efficiencies is where the reaction is carried out by its desired mechanism with very little side reaction. The region A is where the energies are so high that other undesired reactions decrease the quantum efficiency falls off because the photons do not have enough energy to activate the molecules to cause the reaction of interest. One would of course desire to use a source with wavelengths in region B . If this is done, one would expect that the small changes in quantum efficiency with wavelength would not affect the model. This indeed has been shOwn to be the case in several investigations (5, 6). However, often the region B may not be wide and flat; it may be very narrow as in Figure 9. This situation might occur quite frequently since the purpose of going to a photochemical technique is often to provide a method for avoiding interfering reactions with slightly higher activation energies than the reaction of interest. One could of course try to use an essentially monochromatic source over the narrow B region of wavelengths. This would probably be difficult and very inefficient in terms of energy usage. One is then forced to choose between regions A and C or some combination of them. In general one would prefer to operate in region C. Operation in region A destroys quantum efficiency. 4’). 39 \ C. Figure 8. Figure 9. wavelength. A quantum efficiency. '1’). Quantum efficiency vs. wavelength-Tor a typical industrial photochemical system. Quantum efficiency vs. wavel en gth-Tor a, typical actinometric photochemical system. wavelength. X 4O reactant which might otherwise be recycled and also causes a separations problem in that the undesired products formed in region A must be separated. In this investigation a reaction was chosen in which the quantum efficiency versus wavelength curve was essentially like that of Figure 8, but it was investigated with a source having most of its output with wavelengths in the region / corresponding to C. This was done to simulate operations with a quantum efficiency versus wavelength curve like that of Figure 9 with operation in its C region. Over the experimental wavelength region the quantum efficiency was a strongly monotonically decreasing function of wavelength. By combining (multiplying) the curve of Figure 7 with a decreasing §X curve, the wavelength dependence of the monochromatic reaction rate is obtained as a curve of the shape of Figure 10. One would expect that actual reaction rate to be some sort of average of the RX '3 given in Figure 10. There is one final comment that should be made about the qualitative variation of é)‘ with wavelength before a quantitative treatment is given and that is that since the expression for @x given by equation (21) corresponds to a reaction rate expression with the K corresponding to a temperature dependent ”rate constant" and the m-l and n corresponding to non-temperature dependent "orders" of reaction, the K should be the parameter which changes the wavelength while the (m-l) and n parameters should be relatively insensitive to it. (m-l) and n should only change with wavelength if the entire mechanism of the reaction changes. 41 average monochromatic reaction rate , RX wavelength, X Figure 10. Typical curve for the average monochromatic reaction rate as a function of wavelength. 42 In order to quantitatively treat the problem of variations with wavelength the formulation of the monochromatic case must be recast so that the expressions corresponding to equations (42), (43) and (45) are averaged over wavelength. To do this the Lambert law expression given in equation (31) must be restated in terms of intensity per unit wavelength rather than intensity itself. Over any small region of wavelength AX that the intensity is given by I : (1 AK (49) where up X is the intensity per unit wavelength, moles of quanta unit area - unit wavelength ° (The subscript X '3 mean that the variable is a function of wavelength.) Substituting this into equation (31) and dividing out AX which is independent of p , the following is obtained 1 d 2 _ g7 '37) (P GPA) - -H)\ GPA (50) This can be integrated to give the equation corresponding to equation (33) 2 ...s pO HI = GPA = upon) :2- eXpl-Hflp-pofl (51) We now rewrite equations (20) and (21) (over a differential wavelength interval as Rx (ix ap,>\ dX g (52) _ m-l n i, — K, (0p,)\li)\) CA (53) 43 Note that . K lim X _. -——-—— = K (54) where K is the K from equation(21) Using the wavelength limits X1 and X2 shown in Figure 10, one integrates to get for the equation corresponding to equation (34) for R (at the surface p). "2 R=S R dX2g‘ a P x X .lekék 90.x FeXPl-me-pofl (55) 2 o The expression for R is then 3p2 )‘2 — — o R=5 Rde=—3——§S ix up,x{1-8XP[-Hx(P-Poll}d>\ X p -p X o o l * (56) and the expression for P corresponding to equation (40) is "2 Z P— 5), PX dX —ZTT(1-COS¢)pO‘§)\ é)‘ apo,)\{l -exp[-p.)\(p-po)]}dX l * (57) Rmax is now givenxfrom equation (55), setting p = po. 2 Rmax = 5) [ix §X up ,X dX (58) l o 3:‘One might question the order of integrations with respect to p and X . The functions within the integral signs are continuous functions of p for all X in X and X and they are continuous functions of X (or continuous over a finite number of subintervals in X and X for all p greater than p0). These are sufficient condiltions for the double integrals with respect to p and X to exist and for the order of integration to make no difference. 44 The two dimensionless reaction rates are then given imme- diately as X2 2 ‘I w(§(0p (eXPl-H((p-ggldk R po X1 0’ R* : : —2 X (59) max p 2 5X Pm é)‘ GP ,X d)‘ 1 X2 ._ 392 SK Rx0P,X{1-6XP[-Hx(P-90)l}dk _.I. _ R _ 0 1 O 6 R""‘R " 3 3 X ( 0) max p -pO 2 S HX §X 0'p ,X dX X o l Pmax is now given from equation (57) letting p -' 0° . >‘2 2 Pmax — 2w (1 - cos 4)) PO 5X é)‘ up ,X dX (61) 1 o : (62) The dimensionless path length 1* is given as before by equation (46) except that now the absorption coefficient, p. , is some average absorption coefficient, II, which is averaged over wave- length. In the past an input intensity average of IJ. has been used (3,6,12-15), i.e., )‘2 H a dX _ SKI X 90, X ”i _ X2 (63) (1 dX SK 90, X However it will now be shown that this has no fundamental basis and the correct expression for J:- will be derived. As is known, p. , for a monochromatic source, is defined by Lamberts law or in this case by equation (33). Thus one would expect a E which satisfies this equation, namely 2 _. _ PO _ 1p =Ip 7 eXpl-Mp-pofl (64) 0 p where T , —I are averaged over wavelength and come from the p 0 integrated form of equation (49), i. e. , 2 n . T = lim 2 a AX. = S (1 dX (65) p n—ooo i=1 p’ki 1 )‘1 p,X >\Z T = S‘ a dX 66 P )‘1 90. X ( ) Substituting these two equations into equation (64) the following is obtained , _ "2 >2 92 _ _9 -_ - I am dx - I up ,x d) 2 exp[ Hp 90)] (67) X X 0 P 1 L. 1 _ Substituting for a from equation (51) results in p”) 46 2 2 2 2 a :32 ex [-p. (p-— )IdX = 0 dx :QeX ['7 - )] p,XZ P X po p,X 2 Pitppo A]. O p )‘l 0 p (68) Rearranging, X2 g a dX _ ' X1 p0,)\ eXP[Mp-po)] = x2 (69) 5). apo’x exp[-H)\(P'Po)l d)» l T >‘2 5 o. dX — 1 >‘1 p0,)\ p, z in (70) p'po )‘2 IX «1%,, exp[ «(m-pondx l The important difference between the If given by equation (63) and the proper :1.- given by equation (70) is that the latter depends upon the pathlength, p - p0 . Thus 3* the dimensionless length coordinate is given by equation (46) with E instead of p. where [L- is given by equation (70). In this investigation plots were made of R*, R*, and 13* versus 1* taken from experimental data. Note that in order to make these plots from experimental data one must know the absolute values of the “X 's at each X whereas one only requires the relative values of up ,X and §X at each X . If the investigation had dealt O 4 with a chain reaction, the absolute values of these latter two variables 47 would have been required also. The principle objective of this investigation was to see how well one could scale up the three dimensionless variables (especially P*) in the presence of strong variations of up ,X’ ”X , and §X with X . Of particular interest was the variation3 of §X with X which is by far the most difficult to measure experimentally. It is of considerable value to know how accurately a strong variation §X with X must be known in order to scale up a photochemical reactor. The objective of the experimental program about to be reviewed was to investigate equations (59), (60), and (62) during scale-up while taking into account the considerations mentioned above. EXPERIMENTAL W OR K A photograph of the experimental equipment is given in Figure 11. Six conical-shaped reactors were used in the experi- mental part of the study. They were made in sections from acrylic sheets of varying thickness and held together by six rods which were threaded at both ends, the lower ends screwing into a steel base plate and the upper ends securing a steel cover plate and stirrer bearing assembly. The plastic sections were sealed together by applying a small amount of silicone grease between each of the faces. A drawing of a cross—sectional view of one of six reactors is given in Figure 12. Three smaller reactors were made by removing the plexiglass top plate and upper most conical section, changing the stirrer and stirrer bearing and adding a smaller plexiglass top plate. In a similar manner two larger reactors were made by adding conical sections. A table of dimensions for the six reactors is given in Table l. The volume of each reactor was calculated from equation (36) and measured experimentally by filling with a known amount of water. The stirrers for reactors A, B, C, and D were steel coated with acrylic plastic and epoxy resin. tirrers for reactors E and F were polyethylene with steel supports in the shafts. They protruded a distance equal to one third the reactor radius p into the reactor. The blade diameter was approximately one eighth of the reactor radius. The volume and light obstruction of the stirrers were considered negligible in all calculations. Light was assumed 48 49 .2 I ightnin” ... . c s.- Q l i O ... .. ... ......s.?-.a...s..¢.s O 5 reactor c.’ ' O To a ‘I 9 Photograph of the experimental apparatus. . .S ... t .u ”t. h C. o. g . ... he: 8 f a a. S Figure 11. .50 s7/2R\N\&V//\\\ //////////// /////_.//§ a w %§ fix“ %§ 1 x “\“c Ii /§ / “a“ I / . a». / . Cw . // A‘\\ /\\\ , \c / / /§ / \\\x 1 \\ \\\ “N //I / was \ . xxx /// I \\ s//\\\\\\k //A§%////// .4 a \\\\ \Kk I <2:A::> ////////// I 7' SCALE - INCHES \\ Figure 12. Scale drawing for a cross-section view of reactor D. 51 Table 1 Dimensions of the Experimental Reactors Volume Volume Reactor Radius Source Radius Calculated Measured (p) (p ) Geometrically Experimentally 0 cm cm (V) (V cm3 cm3 A 6.477 0.760 19.35 19.5i.5 B 8.786 0.760 48.37 48.5:1: .5 C 11.816 0.760 117.7 118 :t.l D 16.226 0.760 304.8 306 d: .2 E 23.231 0.760 894.6 890 :1: .5 F 33.381 0.760 2654.1 2660 :l: .15 52 to pass only once through the reactor. All external and reflected light was considered negligible in the calculations. The stirrers were turned at moderate speed by a ”Lightnin" Model F electric mixer. Cursory step input mixing tests showed that essentially "perfect" mixing was easily achieved with respect to macrosc0pic concentrations and temperature for all reactors. The stirrer bearings were made of Teflon as were inlet and outlet fittings. These fittings had viton "O" ring seals. The light source used in this work was made by Chicago Miniature Lamp Works. It is their model CM8-807 tungsten filament lamp. It was operated at approximately its nominal voltage of 6 volts where it consumes about . 7 watts of power, approximately 2% of which is emitted as radiation in the region below 700 milli- microns. The dimensions of this lamp and a detailed discussion of its positioning within the reactor is given in Appendix I. Table 2 gives the distribution of relative intensities of this lamp from 300 to 600 millimicrons. This distribution was deter- mined at the Michigan State University Department of Biochemistry using a monochromator and photomultiplier tube. Details of the procedure for this determination are given in Appendix II. The electric circuit for the lamp was designed to keep both the intensity and intensity distribution constant during each run and reproducible in different runs. The tungsten filament of the lamp slowly evaporates with time increasing its resistance. The lamp by itself is thus inherently unstable with respect to its own intensity output with a constant input (either current or voltage). It was U1 {-0 'Table 2 Relative Intensities of the Experimental Light Source VVavelength Intensity VVavelength Intensity VVavelength Intensity (X) (G) (M (a) (X) (a) millimicrons arbitrary millimicrons arbitrary millimicrons arbitrary units units units 300 0.000 405 140 505 927 305 0.000 410 155 510 1000 310 0.331 415 170 515 1040 315 0.991 420 194 520 1120 320 2.64 425 220 525 1190 325 3.98 430 247 530 1270 330 6.67 435 277 535 1360 335 9.41 440 307 540 1460 340 12.6 445 342 545 1540 345 16.6 450 379 550 1640 350 21.5 455 416 555 1750 355 27.8 460 450 560 1880 360 34.1 465 481 565 2000 365 41.1 470 515 570 2140 370 48.5 475 568 575 2350 375 55.5 480 619 580 2610 380 65.8 485 677 585 2950 385 78.2 490 732 590 3270 390 92.7 495 789 595 3540 395 109 500 850 600 3770 400 128 54 assumed that a constant total power consumption within the lamp would give a constant fraction of useful photochemical output and that this output would have a constant intensity distribution even in the presence of an increasing filament resistance. A circuit was then designed to give constant power consumption in the lamp. A drawing of this circuit is shown in Figure 13. The lamp has a resistance of approximately 50 ohms. It can be shown that if the lamp's resistance is exactly equal to the 50 ohm resistance in series with it, the differential change in lamp power consumption with respect to a differential change in lamp resistance is zero. A simple numerical example indicates that a 10% increase in lamp resistance will cause a decrease in power consumption of less than 1/4%. The double throw switch connected to the lamp and a second 50 ohm resistor is used to warm up the power supply and the series 50 ohm resistor before the lamp is switched into the circuit. The power supply was a Hewlet Packard Model 721A 0 to 30 volt supply. 0. 700 watts was the power about which the lamp was held constant. Time was measured electrically with an Atomic Accessories, Inc. Model ET~79A Timer. It read to 1/100 of a minute. The reactant used in this investigation was a 0. 006 molar aqueous solution of potassium ferrioxalate which was 0.1N in H2804. It has been used in the past for actionometric purposes (7, 16-21). However, over the wavelength region of the tungsten source, it has a strongly varying absorption coefficient and quantum efficiency. A table of experimentally determined quantum efficiencies (‘31 55 'vvvv» 500. SOURCE Figure 13. 5051 O ‘1’ CONSTANT VOLTAGE FKHNER SUPPLY Electric circuit for the experimental light source. 56 for this system is given in Table 3. The absorption coefficients as measured by a Beckman DK-ZA spectrophotometer and cells with 0.1, 1, and 10 cm.path lengths are given in Table 4 for the wavelength region of interest, namely 300 to 600 millimicrons. Due to the low conversions, the difference between reactant and product absorption coefficients was negligible. The mechanism for ferrioxalate decomposition is still not well understood but it is known that the ferric iron is reduced stoichiometrically to ferrous iron by the following equation 2[Fe(C 0)]"3 -+2Fe(C 0)+3[c o]‘2+2co 243 24 24 2 This ferrous iron formation is measured by complexing the ferrous iron with 1 - 10 phenanthroline and measuring the amount of orange colored complex formed with a spectrophotometer. The exact procedure was one similar to that of Hatchard and Parker (7). A 10 ml aliquot of reacted solution was emptied into a 25 m1 actinic flask, 3 ml of 0.1% 1-10 phenanthroline solution was added, 5 m1 of buffer solution 0. 6N in sodium acetate and 0. 36N in H2504 was added, and the solution was then diluted to the mark. The solution was allowed to stand for at least 30 minutes to complex. The absorbance of the solution at 510 millimicrons was then measured using a Beckman DK-ZA spectrophotometer. This procedure was carried out in parallel with one using a. 10 m1 aliquot of unreacted solution. The "difference absorbance" was related to concentration of ferrous iron formed by calibrating the spectrophotometer with solutions of known amounts of ferrous iron in a similar way. The 57 Table .3 Experimentally Determined Quantum Efficiences for the Decomposition of Potassium Ferrioxalate Wavelength, X Concentration, C Quantum, é Reference millimicrons mole s/liter efficiency 070 254 0.006 1. 25 254 0. 005 l. 22 19 297 0.006 1. 24 7 313 0.006 1. 24 334 0. 006 l. 23 358 0.006 1.25:}: .01 21 365/6 0.006 1.21 7 365 0. 005 1.20 19 365/6 0.006 1. 26 a: . 03 20 392 0.006 1.131: .01 21 405 0. 006 1.14 7 416 0.006 1.12:1: .02 21 436 0. 006 1.11 7 436 0.005 1. 04 19 436 0.15 1. 01 7 468 . 15 0. 93 7 480 0.15 0.94 7 509 0.15 0. 86 7 546 0.15 0. 15 7 577/9 0.15 o. 013 7 58 Table 4 Absorption Coefficients of 0. 006N Potassium Ferrioxalate in 5 millimicron Intervals from 300 - 600 millimicrons Wave- Absorp- Wave - Absorp- Wave- Absorp- length tion length tion length tion (X) Coeff. (X) Coeff. (X) Coeff. milli- (+1)“ milli- (p. )-l milli- ()J. )-1 microns cm microns cm microns cm 300 37.4 405 2.41 505 0.01509 305 36.3 410 2.11 510 0.01178 310 34.0 415 1.74 515 0.00916 315 30.7 420 1.40 520 0.00709 320 27.7 425 1.12 525 0.00575 325 25.7 430 0.896 530 0.00481 330 22.9 435 0.724 535 0.00414 335 20.7 440 0.541 540 0.00378 340 18. 9 445 0. 443 545 0. 00357 345 17.1 450 0.347 550 0.00350 350 15.5 455 0.263 555 0.00357 360 12.2 465 0.149 560 0.00371 365 10.7 470 0.113 565 0.00375 370 9.31 475 0.0856 570 0.00429 375 7.97 480 0.0640 575 0.00502 380 6.75 485 0.0477 585 0.00535 385 5.69 490 0.0356 590 0.00575 390 4. ‘73 495 0.0251 595 0.00643 395 3.89 500 0.01997 600 0.00651 400 3.20 59 details of this calibration procedure are given in Appendix III. Experimental runs were batch runs. Conversions were very low so that concentration dependence was negligible. The reaction was then practically zeroeth order so that the so-called initial reaction rate was the reaction rate actually measured. The procedure was to sample the reactant (taking a 10 ml aliquot) and fill the cleaned reactor with reactant. The stirrer was turned on and the power supply was also warmed up for at least 10 minutes, using the 50 ohm resistor. The timer was turned on and the lamp was switched into the circuit simultaneously. The run was carried out for the desired length of time. The lamp and timer were then stopped simultaneously. A sample was taken (10 ml aliquot) of the product and analyzed by the method stated previously. Runs were made in a dark room. Room lighting was provided by General Electric BAS, 25 watt, Ruby safelights. All transfer and storage of samples were made either in this type of lighting or in a c orrugated cardboard box. RESUL TS AND DISCU SSION The numerical integrations for R* , R’k , P’l< , E, and E1 were performed over the wavelength region of 300 to 600 millimicrons. No radiation from the lamp was observed below 300 millimicrons. It was felt that reaction would be negligible above 600 millimicrons. This assumption is consistent with the quantum efficiency and absorption coefficient data of Tables 3 and 4. In fact, reaction is negligible at some wavelength lower than 600 millimicrons but there was no a priori way to tell the exact wavelength where this occurred since the exact quantum efficiency distribution was not known. The wavelength region of interest was divided into 60 five millimicron panels and all integrations were by Simpson's one-third rule. Since all quantities calculated were ratios of numerical integrals, the interval width and one-:hird factors would cancel so they were not carried along in the integrations. One of the interesting results was the calculation of the average absorption coefficient, If, by numerically integrating equation (70). Table 5 lists values of E and 1* , the dimensionless path length for various values of p using an initial radius p0 equal to the experimental initial radius of . 760 cm. Obviously I: decreases markedly with pathlength. This means that the wavelengths of lower absorption are weighted more heavily in the larger reactors. The average absorption coefficient, p, seems to be approaching the minimum Il in the 60 61 Table 5 Average Absorption Coefficient and Dimensionless Pathlength as a Function of Reactor Radius for the Experimental Reaction System* Radius (p) Average Absorption Dimensionless cm Coefficient Pathlength (H) cm"1 (1*) 0.7601 0.18946 0. 0000189 0.850 0.146 0.0130 0.900 0.132 0.0181 1. 000 0.112 0.0261 1.100 0.0984 0.0324 1.200 0.0889 0.0376 1.500 0.0711 0.0500 2.000 0.0558 0.0648 2. 500 0.0477 0.0760 3.000 0.0416 0.0852 4.000 0.0343 0.100 5.000 0.0298 0.112 6.000 0.0267 0.123 6.477 A 0.0255 0.127 7.000 0.0244 0.132 8.000 0.0225 0.140 8.786 B 0.0214 0.146 10.000 0.0199 0.155 11.816C 0.0182 0.167 16.226 D 0.0154 0.192 23.231 E 0.0129 0.225 33.381 F 0.0110 0.264 40. 000 0.0102 0.286 50. 000 0. 00934 0.315 75. 000 0.00809 0.375 100. 00 0.00740 0.423 1000. 00 0.00474 0.826 10000. 00 0.00375 0.974 #The letters after certain radii represent the experimental reactors corresponding to those letters. 62 . . .. —l .. . . . distribution (. 00350 cm at 550 mllllmlCI‘OHS) as the radius, p, gets extremely large. For comparison, the incident intensity average absorption 0 oefficient was calculated by numerical integration of equation (63). The. value cf III was found to be . 18953 cm-l. From Table 5 it appears that this is the limiting value of I: as p approaches p0. It appears from these calculations that an incident intensity average absorption coefficient will give a false impression of the true absorption. For instance, over the narrow range of experimental conditions (6. 477 cm to 33. 381 cm) E is considerably lower than I and varies by more than a factor of 2. From Table 5 it can be seen also that the experimental conditions cover less than 1/7 of the domain of the dimensionless pathlength, [1* . Had :1 been used in calculating 1* almost 1/3 of the 11"?“ domain would have been covered (. 550 to . 864). Previous investigations have not considered the variation of quantum efficiency, {A , with wavelength when a polychromatic source is used. The system used in this study was investigated by this method also, assuming that the quantum efficiency is 1):< over the entire domain of wavelengths. Figure 14 gives plots of the quantity é.) o. [l - exp(~ux (p~po))] in equation (56), for X PO» values of ,3 equal to those in the largest and smallest experimental reactor. These are really plots of RX in arbitrary units versus it t is sufficient to assume only that the quantum efficiency is uniform over the entire wavelength domain. 63 600 average monochromatic reaction rate R). arbitrary units 400 REACTOR F 200 REACTOR A wavelength, x , millimicrons Figure 14. Average monochromatic reaction rates vs. wavelength in the largest and smallest experimental—reactors assuming the uniform quantum efficiency model. 64 7x. . There are two interesting features to these curves. The first is the merging of the curves at low wavelengths. This is due to the high absorption which causes the exponential term to be negligible. This in turn causes ilk to be independent of the pathlength, p~pO . The other interesting feature is the strong increase in “RA at high wavelengths. It illustrates the inadequacy of the uniform quantum efficiency assumption. Rx should fall off With Wavelength since quantum efficiency must eventually approach zero. In the largest reactor the simple model predicts even larger R 's at high wavelengths than at low wavelengths. Table 6 gives a summary of the experimentally measured reaction and production rates for the 6 reactors. The data from the specific runs on each reactor are given in Appendix IV. The. ac:uracy of the data for the runs in the smaller reactors where t-.e runs were of short time durations seemed to be better than the data for the runs in the larger reactors. This was due to electrical drift in the source power supply. Power drifted. by as much as $3970 in runs which lasted more than 12 horrs. Intensity output in the lower wavelength region is probably a greater than linear function of power so that errors in the longer runs might he 2 to 3 times greater than this. The worst scatter was observed in Reactor E for which run E-3 was 6. 3% above the average. However the average scatter of the data was less than it 3%. Data or; the smallest reactor scattered less than i 1%. The accuracy of the Beckmann DKZA spectrophotometer was listed as :i: 1%. Total error in the two volumetric measurements is estimated to be :1: O. 4%. Reactor amoom> Summary of Experimental Reaction Data Radius (9) “m \u‘ 6.44? 8.786 11. 816 16.226 23. 231 Ave. Ab- sorptiog coeff. (u) -1 cm 0.0255 0.0214 0.0182 0.0154 0.0129 0.0110 65 Table 6 Dimension- le s s path- length <1 a.) 0.1272 0.1464 0.1672 0.1923 0.2251 0.2639 Volume Ave. W) 3 cm 19.35 48.37 117.7 304. 8 React- Rate R -10 mole cm min 11.4 5.14 2.20 0.855 0.290 0. 1031 Ave. Prod. Ra_1_:e P = R V 10-8 mole min 2.21 2.48 2.59 2.61 Many of the earner runs wze e dis soarded due to corrosion of the stirrers and improper cleaning of the reactor. All of these runs gave excessively high results. The experimental data have been correlated to several models of the quantum efficiency distribution including the simplest model just given. For referer‘ ce purposes, Figure 15 gives graphical representations of these models. The correlations were accomplished in the following way. Dividing the experimental value of the production rate, P , by the predicted value of the dimensionless production rate, pPR , yields a predicted value of the maximum production rate, P . The average of these P 's for the six reactors is max max callfiad the "correlated‘i value of 55 maximum production rate, max . Dividing each experimental value of the production rate, COR P , by the correlated value of the maximum production rate gives , " .N-n-y. ‘ a. (flea—‘61 f” ted" dimensionless production rate, PEOR' The average per cent. of ahsolute dsY/iati on of the P>'-‘ from PlSR gives a COR numerical mess ire of a particular model's ability to correlate the data. As sim; lar method cor rtela ing an Rmsx from experimental A a COR average reaction rates and predicted dimensionless average reaction rates would yield ted .n tical correlations. The correlation for the uniform quantum efficiency model previously mentioned is presented in Table 7. The average per- cent of absolute deviation is 18. 7%. This is of course well outside of the experimental error. Further evidence of the lack of correlation of the simple uniform quantum efficiency model 67 L2 " MODEL 1 0 MODEL 3 quantum efficiency, ‘3. moles reacted eine tein absorbed MODEL 2 , O. 300 400 500 wavelength, X, millimicrons Figure 15. Summary of quantum efficiency models. Table 7 Correlation of Experimental Data for the Uniform Quantum Efficiency Model (Model 1) Reactor Predicted Predicted Correlated P* -P* . . . . . COR PR Dimensionle s s Max1mum Dimensmnle s 3 Pg: Production Rate, Production Production PR 35.‘ * PPR Rate, Pmax Rate, PCOR x 100 Dimensionless PR Dimensionless Dimension- 10’8 moles/min less A 0.1356 17.0 0.170 25.4 B 0.157? 15.7 0.190 20.2 C 0.1819 14.2 0.199 9.4 D 0.2119 12.3 0.201 - 5.2 E 0.2521 9.9 0.199 - 20.8 F 003013 901 0.208 " 3009 Total 78. 2 Total 111. 9 78 2 x 10'8 -8 _- z ' ‘ -- :13.0x10 moles/min max 6 COR 4. " '-=?*-‘ - 1.11.2.9. _ . Avg. (70 ab... dev. OI pCOR -— g — 18. 770 69 is given by the strong positive to negative trend of the deviations as reactor size increases. This model would certainly be inadequate for scale -up since it predicts that the production rate will more than double from the smallest to largest reactor when experimentally an increase of only 23. 5% is observed. The explanation for the poor correlation of the uniform quantum efficiency model is that it weights the higher wavelengths too heavily. Since the higher wavelengths have the higher intensities and the lower absorption coefficients, the uniform quantum efficiency model predicts scale -up ratios which are too high and thus the deviations go from positive to negative as the reactors get larger. The second quantum efficiency model to be correlated is a step down function model. In this model the quantum efficiency is assumed to be 1* from 300 millimicrons up to some critical wave- length. At the critical wavelength the quantum efficiency drops abruptly to zero and remains at zero to the end of the wavelength region at 600 millimicrons. This is the simplest possible refine- ment of the uniform quantum efficiency model. The correlation of this model was attempted at several critical wavelengths. The best correlation was obtained at a critical wavelength of 470 milli- microns and is shown in Table 8. The average absolute deviation of this model is only 2. 0% which is within experimental error. The maximum deviation is only -3. 2%. There seems to be no significant >i< Actually it need only be some constant value. 70 Table 8 Correlation of Experimental Data for the Step Down Function Quantum Efficiency Model with a Critical Wavelength of 470 Millimiorons (Model 2) Reactor Predicted Predicted Correlated P* - P* . . . . . COR PR Dimensmnle s 5 Maximum Dimensmnle s s Pf Prod. Rate Prod. Rate Prod. Rate PR * (P ) sac (PPR) max (PCOR) x 100 PR 10-8 moles min A 0.8462 2.62 0.819 -3.2 B 0. 9003 2. 76 0.915 1.7 C 0.9403 2.76 0.960 2.1 D 0.9698 2.69 0.966 - 0.4 E 0.9889 2.62 0.960 - 3.0 F 0.9971 2.74 1.011 1.4 Total 16.19 [Total] 11.8 — ”"19 X 10-8 — 2 70 x10.8 mole /min max — 6 — ° 5 COR A >1: _ 11.8 _ vg. % abs. dev. of P — —— — 2.0% COR 6 71 trend in the deviations either. The correlations at critical wave- lengths of 465 and 475 millimicrons showed average absolute deviations of 2.1 and 2. 9% respectively, the former tending from positive to negative and the latter tending from negative to positive which might be anticipated from the explanation for the trend in deviations of the uniform quantum efficiency model. The step down function quantum efficiency model shows that a relatively crude model is capable of correlating the experimental data. However apripri prediction of the critical wavelength is likely to prove difficult. The third model for quantum efficiency is merely a curve drawn by eye through the literature data given in Table 3. The numerical values for the quantum efficiencies used in this model are given in Appendix V. The correlation for this model is given in Table 9. The average absolute deviation in this case is 10. 5% which is considerably better than the uniform quantum efficiency model but still short of what might be hoped for. It predicts an increase in production rate of 64% from the smallest to largest reactor which is still remote from the experimental figure of 23. 5%. There is still the positive to negative trend in the deviations, indicating too heavy a weighting of the higher wavelengths. The fourth and last quantum efficiency distribution model was formulated by further consideration of the literature quantum efficiency data and the absorption data given in Table 4. All literature data on quantum efficiency at wavelengths above 436 millimicrons were taken at concentrations of . 15 molar in Correlation of Experimental Data for a Quantum Efficiency Model 72 Table 9 Taken from All Available Literature Data (Model 3) Reactor Predicted Dimensionles s Pr od. Rate (PEER) 0.3660 0.4086 0.4511 0.4978 0. 5525 WHUOUJ3> 0. 6095 Total 32.28 x10- max 6 COR Avg. % abs. dev. of PdOR = 7,—— Predicted Correlated Maximum Dimensionless Prod. Rate Prod. Rate (Pmax) (PgOR) COR 10-8 mole min 6. 04 0. 405 6. 06 0. 446 5. 75 0. 475 5. 25 0. 470 4. 7O 0. 475 4. 48 0. 500 32. 28 ITotall = 5. 46 x10.8 moles/min 62' 7 = 10. 5% 13* -P* COR PR s: PPR 10.7 9.1 5.3 -5.6 - 14.0 - 18.0 62.7 x100 73 ferrioxalate. This is 25 times the concentration used in this investigation. Thus, the quantum efficiency may be significantly affected. Since the quantum efficiency is low in the region above 500 millimicrons, its monochromatic determination is highly sensitive to stray light of lower wavelength. It is believed for these reasons that the curve of Model 3 is too high in the region above 436 millimicrons. From the absorption data in Table 4 it can be seen that absorption of the ferrioxalate reactant reaches a minimum at 500 millimicrons and is lower at 525 millimicrons than at 600 millimicrons. With these thoughts in mind a new quantum efficiency model was estimated. The numerical values of the quantum efficiencies for this model are given in Appendix V. The correlation for this model is given in Table 10. 'This model though drawn from ver‘y'little‘information provides a fairly good correlation of the experimental data. The average absolute deviation is only 3. 4% with a slight positive to negative trend in the deviations. Model 4 predicts a change of production rate of 30. 2% from the smallest to largest reactor which is considerably nearer the experimental value of 23. 5% than the uniform quantum efficiency model (122%) and the model for literature data alone (64%). Even the best correlated step down function model could predict only 17. 9% which is almost as low as Model 4's value is high. Model 4 is assumed then to correspond approximately to the true quantum efficiency distribution curve. 74 Table 10 Correlation of Experimental Data for a Quantum Efficiency Model Taken from Available Literature Data (Model 4) and Absorption Data Reactor Predicted Predicted Dim ens ionle s 3 Maximum Prod. Rate Prod. Rate * (PPR) (Pmax) PR 10-8 mole min A 0. 6094 3. 20 B 0. 7420 3. 34 C 0.7861 3. 29 D 0. 8271 3.16 E 0. 8660 2. 99 F 0’. 8980 3. 04 Total 19. 02 19 02 x 10"8 -8 = ° = 3.17 x 10 max 6 COR . 20 1 x. _ . Avg. % abs. dev. of PCOR - —6 Correlated P* - P* Dimensionless COR* PR Prod. Rate PPR (Pecos) 0.697 1. 0 0.782 5.4 0. 818 4.1 0. 824 0. 0 O. 818 " 5. 5 0. 861 - 4.1 l Totall 20.1 moles/min = 3. 4% x100 75 For comparison purposes the predicted curves R*, R*, and P* versus 1* for quantum efficiency Model 4 corresponding to Figures 2, 3, and 4 for the monochromatic cases are given in Figures 16, 17, and 18. An expanded scale curve of Figure 17 is given in Figure 19 to better show the experimental portion of the curve and how it correlates the experimental data. Figure 19 was correlated by finding an Rmax in a manner similar to that COR for finding P . max COR A comparison of Figures 18 and 19 shows that analysis in terms of P provides a more severe test of the data than does analysis in terms of R. Multiplication by the reactor volume V brings out the scatter in the data. 76 LG 0.8 0.6 dimensionles 8 local reaction rate 0. R.* 0.2 0.0 O. O 0.2 0.4- 0.6 I.O dimensionless pathlength, f * Figure 16. Predicted curve for dimensionless local reaction rate y_s_. dimensionless pathlength using the Model 4 quantum efficiency distribution. 77 [.0 dimensionless average reaction 0.4 rate —>:g R 0.0 0.2 0.4- 0.6 0.8 [.0 dimensionless pathlength, 1* Figure 17. Predicted curve for dimensionless average reaction rate v_s_. dimensionless pathlength using the Model 4 quantum efficiency distribution. 78 [.0 0.8 0.6 dimensionle s 8 production 1: ra e 0.4 13* 0.2 O. 0.0 0.2 0.4- , 05 0.6 [.0 dimensionless pathlength, 1* Figure 18. Predicted curve and correlated values for dimensionless production rate 33' dimensionless pathlength using the Model 4 quantum efficiency distribution. 79 dimensionle ss average reaction rate fifi‘ 0.0 ' ~ , ' v 0.! 0.2 0.3 dimensionless pathlength, 1* Figure 19. Predicted curve and correlated values for dimensionless average reaction rate vs. dimensionless pathlength using the MID—d'el 4 quantum efficiency distribution over the experimental range of variables. CONCLUSIONS Prior to this investigation there has been a trend toward determining the mechanisms of photochemical reactions before attempting scale -up of photochemical reactors. It has been shown by this investigation that knowledge of the photochemical reaction mechanism is not necessarily a requirement for scale -up of a photochemical reactor. It has been shown both theoretically and experimentally that a simple empirical method is adequate for the scale -up of the photochemical reaction source term with respect to all variables associated with the light. The method consists of expressing the local reaction rate by the empirical model R = H l I l i where R is the local photochemical reaction rate. p. is the absorption coefficient of the reaction medium. IT] is the magnitude of the intensity of the radiation. <§ is an empirical parameter determined from the measurement of monochromatic reaction rates. c} is commonly called the quantum efficiency and is the ratio of the moles reacted to einsteins of absorbed quanta. From theoretical considerations of the three major deviations from the Einstein photochemical equivalence law, namely reactions other than the reaction of interest, chain reactions and dark reactions it was found that é could be given by 80 81 where K is the reaction rate constant. m is the exponent on the absorbed intensity and corresponds to the order in thermal reactions. It is unity except for chain reactions. CA is the reactant concentration. n is the reaction order. The method requires three types of data as a function of wavelength: source intensity data, reaction medium absorption data, and small scale monochromatic reaction rate data. This method was tested experimentally using a tungsten source and the decomposition of 0. 006M potassium ferrioxalate in 0.1N sulfuric acid as a reaction. Six cone-shaped reactors were used. Source intensity data and reaction medium data were also taken over the active wavelength region of the source, 300 to 600 millimicrons. From the correlations of the experimental reaction data it was found that the simple empirical model could adequately describe the reaction rate expression for this non-chain reaction. It is generalized from these results that the model could be used for scale up of any photochemical reaction if the empirical parameters (especially quantum efficiency) can be accurately determined as functions of wavelength. Until this investigation, there had been a questionable method used for averaging the absorption coefficient in the Lambert law when a polychromatic source was used. This method was to average 82 the absorption coefficient over incident intensity. This average is given by 9‘2 — ‘8)(1 ”X updx d.\ “‘1 9‘2 SKI GPO,)\ dk where a X is the incident intensity per unit wavelength. 0’ It was shown in this work that the previous method was incorrect. The prOper method of averaging absorption coefficients was derived for the case of spherical geometry with a point source. This average is >‘2 3 ”X 0to .\ d)‘ K o ;_ 1 1n 1 9‘90 3‘2 5). up A 6Xp[-H((p-po)] d\ 1 0 Previously, the standard method for treating quantum efficiency in multiwavelength situations has been to assume that it is unity for all wavelengths. This is, of course, not true. The quantum efficiency falls to zero at both high and low wavelengths. An attempt to correlate the experimental data in terms of a uniform quantum efficiency distribution model showed that it was inadequate for prediction of photoreactor scale ~up for the broad range of reactor sizes used in this study. It was shown that the most probable region of source wavelengths for industrial photochemical reactors is the high wave- length region where the quantum efficiency is falling off. This is 83 exactly the wavelength region where the uniform quantum efficiency distribution assumption will cause the greatest error. From theoretical and experimental results it was shown that the predictions of the uniform quantum efficiency model will predict scaled up production rates which are too high, a more serious error than if they had been predicted too low. From the data correlations of the other quantum efficiency distribution models, it was concluded that even the crudest predictions of the region of high wavelength quantum efficiency drop could give much more confident predictions of photoreactor scale-up. SUGGESTIONS FOR FURTHER WORK There is still no quantitative theoretical model for the quantum efficiency distribution with wavelength. This investigation has shown that such a theoretical model would be very desirable. It would seem that the quantum efficiency at a particular wavelength should be related to the activation energy of the reaction. Assuming there is only one reaction, then the quantum efficiency should be unity for all wavelengths below which the energy of the absorbed photon exceeds the activation energy of the reaction. At higher wavelengths the quantum efficiency should fall below unity. This drop in quantum efficiency should be related to the energy distribution of the absorbing reactant species because only those activated molecules whose total energy is equal to or greater than the activation energy will react. The total energy is the sum of the energy before absorption and the photon energy. The energy distribution of the absorbing reacting species (before absorption) is essentially a Boltzmann distribution and is dependent upon the temperature. If the above ideas are correct and can be put in quantitative terms, then the quantum efficiency distribution with wavelength could be predicted fr om the temperature dependence of the quantum efficiency at a single wavelength The success of model 4 in correlating the experimental data suggests that quantum efficiency is directly related to the absorption coefficient. Any theoretical notions which could quantify the decrease 84 85 in quantum efficiency with decreasing absorption coefficient would also be helpful. APPENDIX I Positioning of the Source Within the Reactor A drawing of the positioning of the lamp is given in Figure 20. The angle <1> in this case is 150. The filament was positioned so that its two edges would just touch the sides of the 300 angle of the "pseudo-apex” of the cone. The lamp was set in place by positioning its top edge a distance of exactly H - L below the t0p face of the lamp base. From geometry H - L was determined to be H—Lz-R—-:-£--L (71) tan 150 The D 2 2R, d = Zr, and L were measured using a graduated magnifier. They were respectively . 230, . 080, and .150 inches. Thus .230 _.080 2 2. H'L ’ .2679?) " “150 _ ._115—.040 ,: _ .075 H'L “ .23795 "'1“0”.26795 " “150 H-L = .280~.150= .130inohes The top face of the lamp was placed .130 :1: . 001 inches below the top face of the lamp base. The lamp was not a perfect portion of a sphere where it touched the reaction medium. The radius of the source pO could be well approximated by the distance from the ”pseudo-apex" to the top face of the lamp. This is given geometrically by 86 L Vi ,. 1 \ I \ ¢ ”__.—- “ Q = .26705 88 -—;£——3 +-L (72) tan 15 an 2 -+150==.149-+.150==.299inohes .760 cni APPENDIX 11 Source C alibrati on The relative intensities of the light coming from the tungsten lamp were measured using a Bausch and Lamb monochromator and an RCA 1P28 photomultiplier tube connected to an amplifier. The lamp was powered by the power supply used in the reactor experi- ments. The light from the lamp was directed into the monochromator which focused it into a beam which was refracted. The refracted segments were approximately 1 millimicron wide. These portions of light were shone on the photomultiplier tube which conducted current in direct proportion to the amount of energy falling upon it. The prOportionality factor of course varied with wavelength. The energy from the beam segment is given by .. E. E — (1k AX h K (73) where EX is energy of the beam segment in erg/sec a is the intensity per unit wavelength in photons/mp. -sec X Al is the wavelength interval of the segment in mp. h is Planck's constants, .6. 62 X 10-27 erg-sec/photon c is the speed of light, 3. 00 X 101-7 mil/sec X is the wavelength, mu The reading was current which was proportional to the energy or in equation form 00 S E (74) 89 90 S x is called the relative sensitivity and can be obtained from data given by the manufacturer of the photomultiplier tube (22). Equations (73) and (74) can be solved for ax to yield .. ‘lx k SxAkhc (1 (75) Since AX, h, and C are constants which are independent of X and since only relative (not absolute) values of 0x are required for the calculations, the following equation can be used to calculate ax 0. = —— (76) Table 11 gives the results of these calculations. 91 Table 11 Quantities Used in Calculating the Relative Intensity Wavelength (X ) millimicrons 300 305 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 420 (5,) 0.880 0.913 0.936 O. 953 0.970 0.981 0.990 0.997 1.000 0.998 0.993 0.984 0.972 0.959 0.947 0.932 0.919 0.906 0.891 0.877 0.860 0.844 0.828 0. 812 0.797 Sensitivity dimensionless Distribution of the Tungsten Source Current (2,) ampheres 0.000 0.000 0.001 0.003 0.008 0.012 0.020 0.028 0.037 0.048 0.061 0.077 0.092 0.108 0.124 0.138 0.159 0.184 0.212 0.243 0.276 0.291 0.312 0.333 0.368 Intensity per unit Wavelength (ax) . arbitrary units 0. 000 0.000 0.331 0.991 2.64 3.98 6.67 9.41 12.6 16.6 21. 5 27. 8 34.1 41.1 48. 5 55. 5 65. 8 78.2 92.7 109 128 140 155 170 194 Wavelength (M millimic r on s 425 430 435 440 445 450 455 460 465 470 475 480 485 490 495 500 505 510 515 520 525 530 535 540 545 550 555 92 Table 11 (continued Sensitivity (5,) dimensionless 0.780 0.766 0. 750 0.731 0.715 0.697 0.680 0.664 0.647 0.631 0.613 0. 598 0. 579 0. 562 0. 547 0. 530 0. 508 0.488 0.469 0.450 0. 432 0.414 0.392 0.370 0.351 0.329 0.305 Current (2,) ampheres 0.404 0.440 0.477 0.510 0.549 0.588 0.621 0.650 0.670 0.692 0.733 0.771 0.808 0.840 0.872 0.901 0.932 0.957 0.968 0.969 0.981 0.994 1.000 0.997 0.990 0.978 0.960 Intensity per unit Wavelength (a,) arbitrary units 220 247 277 307 342 379 416 450 481 515 568 619 677 732 789 850 927 1000 1040 1120 1190 1270 1360 1460 1540 1640 1750 Wavelength (X ) millimicr ons 560 565 570 575 580 585 590 595 600 93 Table 11 (continued) Sensitivity (s,) dimensionle s 3 0.281 0.261 0.240 0.216 0.193 0.167 0.145 0.123 0.100 Current (2,) ampheres 0.941 0.923 0.902 0.883 0.867 0.842 0.803 0.732 0.629 Intensity per unit Wavelength ((1,) arbitrary units 1880 2000 2140 23 50 2610 2950 3270 3540 3770 APPENDIX III Calibration of the Spectrophotometer for Ferrous Iron A procedure similar to that used to analyze the reaction runs was used to calibrate the Beckman DK-2A spectrophotometer for ferrous iron. The following solutions were made: (A) freshly prepared.250 x 10'3M Feso4 which is . 1N H2 2 (C) . 1% 1 - 10 phenanthroline, (D) buffer solution . 6N NaAc and so4 (B) .1N'H so 4’ . 36N HZSO Successive aliquots of solution A in 1 ml increments 4 0 from 0 to 10 ml were added to 25 m1 actinic flasks. A complimentary amount of solution B was added to make the acidity equivalent to SO 10 m1 of . 1N H .3 m1 of solution C and 5 m1 of solution D 2 4 were added to each flask. Finally, each flask was filled to the mark. After allowing the solutions to stand for at least 30 minutes, their absorbance was measured at 510mp. . The difference between the absorbance of a solution and the abs orbance of the solution containing no FeSO4 is proportional to the concentration of FeSO4 by Beer's law. This relation can be written AA = e C (77) where AA is the difference absorbance in a.u. E is the molar extinction coefficient in a. u./mole/liter C is the concentration of F‘eSO4 in mole/liter Table 12 shows the results of the calibration runs. The above result compares well with the values 1.105 x 104 of Parker (16) and 1.11 x 104 of Smith, McCurdy and Diehl (23). 94 95 Table 12 Results of Spectrophotometer Calibration Runs Cone. of Difference e _ L FeSO4 in Absorbance - c Flask, (AA) 105 a.u. -liter moles/liter a. u. mole 1x10"5 0.115 0.115.1105 -5 5 2x10 0.222 0.111 x10 -5 5 3 x10 0.338 0.113x10 4x10'5 0.457 0.114x105 55.10"5 0.554 0.111x105 6 5110'5 0.676 0.113 x105 7 x10"5 0.788 0.113 x105 8x10“5 0.906 0.114;..105 9x10'5 1.015 0.113x105 10x10'5 1.132 0.11311105 Total 1.130x105 (gt) _ 1.13031105 _113X104 a.u. _ o avg. _ 10 _ ' moles/liter >l< The abbreviation “a. u.' stands for absorbance units. 96 Let (3 be the factor which when multiplied by the difference absorbance of the treated product solutions will convert it to moles of Fe+2 formed per cm3. -3 liter sol. 25 cm3 sol. 3 l mole/liter sol. 10 1.13x104 a.u. om3 sol. 10 cm '53 I (78) -7 mole 51cm3 a. u. 2.21 x 10 7G H APPENDIX IV Reaction Data Table 13 Data on the A Reactor Difference Abs orbance Time Run AA, a.u. t, min A—A- 35‘" t min A-15 0.261 50 5.22 x 10'3 A-16 0.309 60 5.15 x10”3 A-l7 0.206 40 5.15 x10.-3 A-18 0.103 20 5.15:.10'3 A-l9 0.155 30 5.16 x 10'3 3 Total 25.83 x 10' (952. : 25’83 x10.3 : 5.17x10-3 a.u./min t avg. 5 3 10 mole E = 8 (513—5ng = (2.21x10'7)(5.17x10' )=11.4x10‘ cm -min P = iv = (11.45.1040 —r—n—°—13—— ) (19.35 cm3) = 3°21X10-8 $11: cm -min * . . . The abbrewation "a. u. " stands for absorbance units. 97 Run B-7 B-8 B-9 B-10 B-ll P = ’fiv = (5.14x10‘ 98 Table 14 Data on the B Reactor 10 mole cm -min mole Difference Absorbance Time AA; a.u. AA, a.u. t, min t ’ min 0.273 120 2.28 x10-3 0.225 100 2.25 x10”3 0.183 80 2.29 x10".3 0.141 60 2.35 x10-3 0.097 40 2.42 x10-3 11 59 10’3 -3 = ' 5; : 2.32x10 a.u./min —) = (2.21x10‘7)(2.32x10'3) = 5.14 x10-10 t avg. cm )(48.37 cm3) = 2.48 x10.8 mole/min 99 Table 15 Data on the C Reactor Run Difference Absorbance Time _A_A_ a.u. AA, a.u. t, min t ’ min c-3 0.202 200 1.010 x10"3 C-6 0.304 305 0.996 x10-3 C-8 0.153 155 0.986 x10“.3 o-9 0.110 110 1.000x10'3 c—10 0.249 250 0.99? x10.-3 -3 4.989 x 10 (é‘é‘) ‘4.98C)XI0-3 “0 997x10-3a u/min t avg. “ 5 ‘ ° ° ' ‘ = “Agile. =(2.21x10‘7)(0.997 x10-3) = 2.20 x10"lo & vg. cm -min — -10 3 -8 . P=RV=(2.20x10 )(117.7cm ):2.59x10 mole/min 100 Table 16 Data on the D Reactor Difference Absorbance Time AA .3. u Run AA, a.u. t, min —, t min 13-3 0. 342 900 0. 380 x10“3 D-4 0.181 450 0.402 x 10'3 13-5 0.238 600 0.397 x10-3 D-6 0.462 1210 0.382 x10-3 13-7 0.279 750 0.372 x10.-3 -3 Total 1.933 x 10 AA 1 933x10‘3 -3 (— = ' :0.386x10 a.u./min t avg. 5 'fi 2 (3 (9%)an -_- (2.21x10‘7)(0.386x10'3) = 0.855 x10-10_____I‘I_}_9_1_§___ cm -min 10 mole P 2 RV = (0. 855x10- )(304. 8 cm3) = 2.61x10-8 mole/min cm -min 101 Table 17 Data on the E Reactor Difference Absorbance Time AA Run AA, a. u. t, min ——-, 1.11.; t min E-3 0.405 2900 0.1397 x10-3 13-4 0.343 2650 0.1294 x10.-3 13-5 0.341 2780 0.1249 x10-3 -3 Total 0.393 x 10 AA 0 393x10‘3 -3 (— = ° = 0.131 x 10 a.u./min t avg. 3 fi = 8 (974‘- : (2.21x10'7)(0.131x10'3) = 0.290 x 10'10 49313— avg. cm -min p = fiv = (0.290;..10‘10 —r-29-1-§—)(894.6 cm3) = 2. 59 x10.8 mole/min cm -min 102 Table 18 Data on the F Reactor Difference Absorbance Time Run AA, a. u. F-6 0. 201 t, min 4310 AA _ -3 . (7)81ng _ 0. 0466 x 10 a.u./min 5' = 8 (9%.... P = fiv = (0.103 X (2.21 x10—7)(0. 0466x10' 0.103x.10‘10 10 -10 3.) )(2654.1) = 2.73 x 10'8 AA a. u. t’ min 0.0466 x10- mole/cm3 -min mole/min 3 APPENDDCV Quantum Efficiency Data for Models 3 and 4 Table 19 Numerical Quantum Efficiency Data for Model 3 Wave - Quantum Wave - Quantum Wave - Quantum length Efficiency length Efficiency length Efficiency mi)lli- molgs reacted/ mi>lli- mole§s reacted/ mi)lli- moles reacted/ microns mole quanta microns mole quanta microns mole quanta absorbed absorbed absorbed 300 1.251 405 1.131 505 0.836 305 1.246 410 1.124 510 0.790 310 1.241 415 1.116 515 0.730 315 1.236 420 1.107 520 0.661 320 12.30 425 1.099 525 0.580 325 1.225 430 1.091 530 0.481 330 1.220 435 1.081 535 0.366 335 1.214 440 1.069 540 0.245 340 1.208 445 1.057 545 0.172 345 0.203 450 1.044 550 0.127 350 1.197 455 1.032 555 0.096 355 1.191 460 1.018 560 0.070 360 1.185 465 1.003 565 0.049 365 1.179 470 0.991 570 0.031 370 1.172 475 0.976 575 0.015 375 1.167 480 0.960 580 0.006 380 1.161 485 0.941 585 0.000 385 1.154 490 0.922 590 0.000 390 1.148 495 0.899 595 0.000 395 1.142 500 0.871 600 0.000 400 1.137 103 104 Table 20 Numerical Quantum Efficiency Data for Model 4 Wave - Quantum Wave - Quantum Wave - Quantum length Efficiency length Efficiency length Efficiency mi)lli- molg reacted/ milli- mole reacted/ milli- mole reacted/ microns mole quanta microns mole quanta microns mole quanta absorbed absorbed absorbed 300 1.251 405 1.131 505 0.121 305 1.246 410 1.124 510 0.097 310 1.241 415 1.116 515 0.076 315 1.236 420 1.107 520 0.061 320 1.230 425 1.095 525 0.048 325 1.225 430 1.081 530 0.039 330 1.220 435 1.062 535 0.023 335 1.214 440 1.034 540 0.013 340 1.208 445 1. 003 545 0.006 345 1.203 450 0.967 550 0.000 350 1.197 455 0.916 555 0.000 355 1.191 460 0.869 560 0.000 360 1.185 465 0.776 565 0.000 365 1.179 470 0.680 570 0.000 370 1.172 475 0.558 575 0.000 375 1.167 480 0.423 580 0.000 380 1.161 485 0.326 585 0.000 385 1.154 490 0.245 590 0.000 390 1.148 495 0.187 595 0.000 395 1.142 500 0.150 600 0.000 400 1.137 b.35‘h-1H7mcjouo H 1* NOMENCLATURE English Letters Absorbance, absorbance units (a. 11.) Positive number, cm Speed of light, 3. 00 x1017 mil/sec Concentration, moles/cm3 Distance variable, inches Distance variable, inches Energy, ergs Function Volumetric flow rate, cm3/min Planck's constant, 6.62 x 10.27 erg-sec/photon Distance variable, inches Current, amperes Intensity, einsteins/cmZ-sec Photochemical reaction rate constant, (moles formed) (einsteins absorbed)l -m/(mole/liter)n(cm3) Dimensionless pathlength Distance variable, inches Exponent of intensity in reaction rate expression, dimensionless Order of reaction, dimensionless Production rate, moles/min Distance variable, inches Reaction rate, mole s/cm3 -min Distanc e variable, inche s 105 >l< 106 Relative sensitivity, dimensionless Time, min Volume, cm Greek Letters Intensity per unit wavelength, einsteins/cmZ-sec -mp. Conversion factor (see Appendix III), 2. 2.1 x 10.7 (moles formed/ cm3)/a. u. Molar extinction coefficient, a. u. /(mole/ liter) Angular spherical coordinate, radians Wavelength, millimicrons (mu) Absorption coefficient, cm-1 3 . 14159 Radial spherical coordinate, om Optical pathlength, dimensionless Angular spherical coordinate, radians Quantum efficiency, moles reacted/einsteins absorbed Superscripts Average over the reactor Dimensionless Vector avg A,B,C chain COR i I inerts max 0 PR r ri I'P rs SS 107 Subscripts Average Components in a reaction Consumption of reactant Chain reaction Correlated Formation of product Initial, integer variable Incident Inerts Maximum Incident Predicted Reaction Reaction of interest Parallel reaction Series reaction Steady state Thermal Function of wavelength Radial c omponent BIBLIOGRAPHY 1) Doede, C. M. and Walker, C. A., Chem. Eng., 62, 159 (Feb. 1959). 2) Marcus, R. J., Kent, J. A., and Schenk, G. 0., Ind. Eng. Chem., 24, 20 (1962). 3) Harris, P. R., M. S. Thesis, Northwestern University, Evanston, 1964. 4) Cassano, A. 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