studies of light-sensitized reactions have Kinetic continued at a sustained rate many decades. OVQ'
1 I I
Such systems provide a controllable environment for investigating mechanimns and products of chain reactions. More recently commercial applications, particularly chlori~tions, have stimulated interat in designing efficient reactors. Empiricism, experience, and art have been the tools for projecting laboratory apparatus and procedures to large-scale operations. Lad of knowledge of pazinent design parametera and methods and the almost complete absencc of quantitative rate data arc major obstacles to progress. However, in the last few years there have been several studies directed specificalIy toward the design problem. Interest is increasing in the photochemical route for p m s requiring a high selectivity-for example, the production of primary chlorides of paraffin hydrocarbons for the manufacture of biodegradable detergents. Therefore, it is appropriate to s 4 the state of the art. The plan in this paper is to review critically the accomplishments in engineering methods, then ddine in mathematical language the design problem, discuss solutions for special cases, and finally comment on unique design aspects. The approach will be to adopt the conventional reactor as a reference so that the comparison with familiar probleas is clear. This procedure will require frequent excursions off the main path to inquire about complex rate equations for chaii 18
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
kinetics, radiation enugy conservation, wall mctions, r a d i i gradients, wall deposits, and effects of wavelength distribution, all of which can be important in photreactor design.
PHOTOCHEMICAL AND CONVENTIONAL REACTORS The unique feature of a photoprocess is its selectivity, owing to the paaaibility ofraising the electronic energy of a molecule to a specific level by the p r o p choice of frequency. This often can be done without disturbing translational and rotational lev+an impwibility by thermal excitation. An illustration is the chlorination of the methyl group of toluene without noticeable reaction in the ring ( 4 7 4 . When combined with a low operating temperature, characteristic of most l i h t initiated chain reactions, the high selectivity offem an attraetive p'ocedure. This type of reaction is currently the most im-t industrial application of photoreactors. The selectivity of photoreactions offers unique possibilities. First, the photoprocess may make feasible the production of a product which would be thermodynamically impossible by thmnal means. Thus, heating to obtain the required energy for an exothermic reaction may reduce the equilibrium conversion for the desired product to an insignificant level. Another pwibility is that a reaction may taLe place photochemically in the liquid phase, while to obtain the required energy by thermal means would reqquire gaseous
A A CNical review of h e accomplishme& in engineering mehods in h e photochemical processes requirins a high SelecfivitY in We reacfion, d e w o n of the design problems of a phofochemical reactor, and sohfionsforspecial casesof economicinferesf
k E. CASSANO P.L SILVESTON
conditions. This introduces the disadvantages of lower production rate per unit volume, higher temperatures, and more expensive materials of construction. The possibility of reaching a given energy l e d with the proper wavelength of radiation has other advantages. If the energy is increased thermally, all the intermediate levels are t r a n s v d , and at m e of these stages other reactions can occur. I n other words, kinetic considerations, as well as thermodynamic ones, may forbid certain’pmducts by the thermal route which photochemically are possible. Despite these advantages, photoreactors have been employed industrially only when the reactions are impossible or severely uneconomical by thermal or conventional catalytic paths. Until recently the few connnercial photoprocesses have been developed d e r expensive laboratory and pilot plant activities (34-4). Reasons for this are the lack of suitable reactor models and design methods, as well as lack of quantitative information about the pertinent physical and chemical parameters. As a single illustration, no direct methods have been found, and hence no results are available, for obtaining diffusivities of activated chemical species, h e radicals, and atoms resulting from energy absorption. Hence, little can be said about the significance of diffusional Effects in many types of photoreactors. Dimensional analysis, of value for conventional reactors, will be Seen later to be of little assistance when applied to the chain sinetics characteristic of most photoprocesses.
Other reasons are based on size limitations and general construction difficulties. The performance of a photoreactor is always coupled with the behavior of the radiation s o u r c e t h e lamp. The type of light and the configuration of the lamp-reactor system affect the reactor design so strongly that independent consideration is not possible. With chain reactions, the potential for explosions, instability, wall deposits with associated reduction in yields, and Severe inhibition are factors that have mitigated against the photo route. The requirement for using glass or silica equipment imposes size limitations and breakage costs. Lifetime limits and control problems of the light source are unique operating problems. Finally, the price paid for selectivity in terms of energy can be high since the energy absorbed by the reaction is always a small part of the power consumption of the lamp. The previous paragraphs have referred to some of the design problems, as distinct from construction, materials, and operating problems in photoreactors. A more complete accounting of the design questions would include the following: S i g n i b c e of wall reactions Diffiuivities of activated species Optimal wavelength range Treatment of the dark reaction Evaluation of the light path for heterogeneous systems Activation energy of the reactions controlling the overall rate VOL 5 9
Applicability of the stationary-state assumption in formulating the rate of reaction Existence of an inner filtering effect which would change the rate of reaction formulation. Inner filtering effect refers to the absorption of light by other substances than those which undergo the desired light-catalyzed dissociation Complexity in solving the conservation equations describing the reactor performance. Analytical solution for complex reaction systems, even relatively simple ones, appears unlikely and machine calculation procedures would tax the capabilities of most present-day computers
HISTORY OF PHOTOREACTOR DESIGN New journals and monographs ( 7 2 3 , 74B) have recently appeared specializing in the chemical characteristics of photoprocesses. These publications and the extensive past and current literature on photochemistry will not be reviewed since the objective here is the design aspect. T o our knowledge, the first contribution is that of Bhagwat and Dhar (7A) who studied the effects of stirring on the rate of reaction. By applying Lambert's law with a constant attenuation coefficient to a flat plate reactor, they obtained for the light absorption, la, the expression
If the rate of reaction fl is proportional to the n-power of the light absorbed per unit length of light path,
For weak-absorption nearly transparent media, stirring has no effect on the activation energy With the exception of papers on practical operating conditions for certain commercial photoreactors, there were no further publications on photoreactor design until 1955. The reviews by Doede and Walker (76A) and Marcus, Kent, and Schenk (34A) started the second generation of design studies. These works were useful descriptions of the industrial importance of photoreactors, and the former provided information on lamp availability and how to arrange the equipment for a large-scale photochemical process. The first quantitative design study (34 5A) was applied to a difficult experimental system : the reaction between hydrogen sulfide and n-octene-1. Severe density gradients and a significant inner filtering effect, due to impurities in the reactants, thwarted a thorough analysis of the data. The pioneer contributions were the reactor configuration and ellipsoidal reflector with lamp and reactor at the foci. In 1958 Gaertner and Kent (2OA) used the well known oxalic acid-uranyl nitrate photodecomposition to study the Roscoe-Bunsen law in a continuous, annular reactor operating at low conversions. Since the reaction is zero order with respect to concentrations, general conclusions about photoreactor design could not be reached. The theoretical treatment in the paper is not entirely rigorous since the form of Lamberts' law applicable for plane geometry was used. However, most of the assumptions were valid and their data checked the derived relationship x =
This equation holds for the whole reactor, if it is completely mixed. If mixing is incomplete, the authors proposed (3)
If the light absorption is very high, ratio of the rates is:
(4) For low absorption, e - p s
+ (1 - p s ) ,
n _ -1 0'
Experimental observations and the above equations lead to the following conclusions, still valid after 30 years :
If the rate is first order with respect to light absorbed (n = l), there is no effect of stirring on the rate (comparison of Equations 2 and 3) If n is less than unity, stirring increases the average rate For strongly absorbing media, an increase in stirring results in a decrease in apparent activation energy for n # 1 20
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
kI,[Rzp - 1 - (Rip - I ) e - ~ ( ~ p - ~ 1 ) ] t CoAw'
showing that the conversion was directly proportional to the mean residence time in the lighted section of the reactor. This contribution showed that the oxalic acid-uranyl nitrate system could be used in a flow reactor to measure the light intensity at the inside wall of the reactor. The annular reactor, with a coolant flowing through an inner tube and combined with an elliptical reflector, provides a method of adjusting the reactor temperature independently of the heating effects of the lamp and the heat of reaction. We have found ( 8 A ) the oxalic acid-uranyl sulfate system a satisfactory method for evaluating the intensity in a cylindrical reactor. However, the method is subject to improvement. Foraboschi (78A) applied dimensional analysis to the photoreaction aA 4 bB under steady-state conditions with laminar flow and monochromatic light. Additional conditions were : Primary quantum yield equal to unity Negligible diffusion iXo reflection of light at the reactor walls A rate equal to the form D = k(al,,,C,)"
where k is a function of CA, C ,, and the specific rate constants for the intermediate steps. Also it is supposed that k at inlet conditions is known By applying the Buckingham theorem to mass and energy conservation equations and' the stoichiometric relationships, complicated equations were developed for relating the conversion to various dimensionless groups. Even for the scaleup problem for the same reaction and with the four listed assumptions, at least seven dimensionless groups must remain constant to maintain similitude. Hence, dimensional analysis does not appear to be a valuable tool for scaleup of photoreactors involving complex kinetics (for example, chain reactions). Schechter and Wissler (45A) analyzed theoretically an isothermal, laminar flow, tubular reactor under the following assumptions : Negligible axial difFusion Constant extinction coefficient (normally not valid) Reaction first order in light intensity and concentration-Le., first order with respect to the light absorption per unit volume, with p proportional to C. This means
Under these conditions, the radiation and mass conservation equations and boundary conditions can be formulated as a Sturm-Liouville problem. The solution gave the concentration ratio C A / C A , as a function of radial r and axial z positions. This was the first rigorous approach to reactor design. An additional contribution was the correct formulation of the boundary condition for mass conservation at the axis of the reactor where the light intensity becomes infinite and the radius zero. However, industrially important photoreactions-for example, chlorinations-cannot be characterized by the simple form of the rate expression given by Equation 8. Nevertheless the published graphs of the solution can provide a qualitatively correct view for some simple reaction systems with low optical densities and without wall effects. Huff and Walker (29A) studied experimentally the vapor phase photochlorination of chloroform in a tubular reactor, particularly the effects of flow rate, concentrations, reactor diameter and length, and radiation source. Valuable engineering contributions were : The first attempt at a quantitative study of a chain reaction in a continuous reactor The use of a summation procedure as an approximation for polychromatic light The successful application of the oxalic acid decomposition reaction for measuring the light intensity at the inner wall of a tubular reactor Analysis of the data to establish the kinetics of the reactions was difficult because of the relatively large conversions. The results suggest that a change from laminar to turbulent flow affected the relative importance of homogeneous and heterogeneous deactivation
reactions. This paper is a valuable source of information for those who plan to initiate engineering studies of photoreactors. Harris and Dranoff (23A, 24A) formulated the problem of completely mixed photoreactors under the restrictions : Radial light Constant extinction coefficient (follows from condition of complete mixing) Rate equation of the form
(9) where p is a function of wavelength and @ depends upon wavelength, concentration, and light intensity Monochromatic light (or 9 independent of A) For an annular reactor with the lamp at the axis, the average rate of reaction was derived to be : =
9 [ 2 TL RJ,][1 -
This result was compared with experimental measurements for the photodecomposition of chloroplatinic acid in very dilute aqueous solutions (a nonchain reaction). Data were obtained for both batch and flow operation. The chief objective was to compare predicted and experimental values of the ratio of rates, a, for two values of RP. If the reaction is homogeneous, good agreement simply verifies Lambert's law. Dolan, Dimon, and Dranoff (75A, 77A) applied dimensional analysis to the following situation : Well developed flow, laminar or turbulent, in a cylindrical photoreactor irradiated from the outside radially with monochromatic light Dilute, incompressible reaction fluid Isothermal, steady-state operation I t is assumed that the overall rate is
where NAhf is the factor necessary to convert the energy E to intensity I,. The overall quantum yield 9 is generally a complicated function of wavelength, light intensity, and reactants and products concentrations. The authors conclude that the conversion could be correlated in terms of the following dimensionless groups : x =
~[(NR~Ns~NL~), A , Na1
where the first three are the Reynolds, Schmidt, and length dimensionless groups and the absorption and reaction numbers are :
and I' are evaluated at I = I, and C = CO. Equation 12 was checked with experimental data on the liquid phase hydrolysis of chloroplatinic acid. Precise
VOL. 5 9
JANUARY 1 9 6 7
measurements were made for the effects of flow rate, reactor length, and diameter on the conversion, and the results were correlated by plotting conversion us. NRe (RIL). Since this latter parameter is inversely proportional to the mean residence time, it is not surprising that a correlation was obtained. For simple nonchain reactions and for monochromatic light the results can be useful. For more realistic systems the problem is much more complicated, as suggested by Foraboschi ( I 8 A ) . Vul’fson (50A) considered the photoreaction of a gaseous reactant which absorbs light. The rate of reaction was assumed to be controlled by the rate of absorption from the gas phase. Transient and steadystate operation of a recycle reactor is analyzed. No new theoretical concepts are introduced, and the description of the photoprocess appears too simple for the results to be of general use. A careful analysis of mixing effects in isothermal photoreactors by Hill and Felder (27A) provides much useful information for design. Recently an experimental and theoretical study of the photochlorination of propane was carried out (8A) in a differentially operated, cylindrical, flow reactor with elliptical reflector and polychromatic lamp. An attempt was made to express the design problem in general form and then achieve, experimentally, simplifications which permit solution of the design equations. A key part of the analysis was the formulation of the overall rate equation in terms of the kinetic constants for the individual steps. A first-order deactivation of the propyl radical was assumed to be the controlling termination process and led to the following equation for the average rate over the radiation path:
where K is a combination of kinetic constants for the initiation, propagation, and termination steps, and 0 is the transmission coefficient for the filter solution surrounding the reactor tube. Data taken for various flow rates (always in the laminar range), concentrations, reactor lengths, and light intensities led to absolute, numerical values for K. While this study did involve chain kinetics, the reactor operating conditions were not practical and much work needs to be done to investigate effects of wall deposits, wavelength distribution, and significance of radial gradients. A recent symposium ( I A ) on photochemical processes included primarily papers concerned with other types of radiation than visible light and with biological reactions. An exception was the work of Walker and Baginski (57A) previously described. Finally Jacob and Dranoff (30A) have refined their earlier experimental work (244). The verification of the method for radial scaleup of completely mixed reactors with nonchain kinetics appears sound. The assumption that the quantum yield is independent of wavelength introduces little error because of the short range of X used in the study, but for other conditions the effect could be significant. 22
INDUSTRIAL A N D ENGINEERING CHEMISTRY
I n summary, engineering research on photoreactors is in its infancy. The initial contributions should serve as a stimulus for the intensive effort needed to develop suitable design and scale-up methods. We believe that the conservation equations combined with proper boundary conditions, particularly for heterogeneous processes, are the proper starting point for a rational design.
DESIGN EQUATIONS In addition to momentum, mass, and thermal energy equations, an expression for the conservation of light energy (radiation equation) is necessary to establish the conversion in a photoreactor. The radiation equation is a unique feature of photoreactors, and the mass balance expressions are more complicated than for conventional reactors because of the complex and radiation-dependent kinetics. The momentum expression is the same as for conventional reactors and need not be repeated. Mass Conservation Equations
The rate of reaction term in the mass balance is a function of composition, temperature, pressure, and volumetric rate of light absorption, la. Therefore, a series of differential equations is required covering the overlapping spectra for the emission of the lamp and the absorption of the reactants. This superposition establishes the limits, X I and XZ, between which reaction can occur. A second problem arises because, for all but the most simple photoreactions, the kinetics cannot be represented by an overall stoichiometric relationship. A detailed mechanism, or simplified sequence of steps, is needed to describe the rate. This means that, in general, a mass balance including diffusion terms is needed for each independent chemical species. Even with a set of differential mass balances to account for the spectral distribution of the source-absorbent system and the various species, the description is not complete if wall reactions are significant. For constant transport properties, monochromatic light, and incompressible flow, the mass balance may be written :
where the molecular and turbulent contributions to the diffusivity are assumed to be additive and is the local rate of production of species i. Thermal Energy Equation
Radiation will be used to describe that part of the energy absorbed which causes reaction. The term thermal energy will denote all other forms. The increase in internal energy due to radiation arid the accompanying temperature rise are, in most cases, the least significant effect of the light. Hence, the temperature in the reactor can be described by a thermal energy balance. This procedure permits the radiation balance, necessary even for isothermal operation, to be treated separately. For nonisothermal conditions, the effects
of temperature on the values of the parameters in the radiation balance must be taken into account. At present this is a somewhat academic question because of the lack of information about the light absorption process. The thermal energy balance may be written for any reactor geometry in vector form as follows: V(energy flux)
where AH is the heat of reaction and E the internal energy. The summation would include all forms of energy transfer. Radiation Equation
I n general, there will be one radiation balance for each wavelength and each direction of radiation, as expressed by
V . I m , x = -Pxpx,ml
where m identifies a direction and the attenuation coefficient, pk, depends upon the wavelength. For unidirectional, monochromatic radiation this reduces to the vectorial form of Lambert's law, applicable to any reactor geometry :
The product 1.1111is the volumetric absorption rate, l a ; 1.1 is conveniently divided into two parts, the first refers to the light absorbed by the molecule that is activated and the second, 1.10, is the constant ground extinction coefficient corresponding to the energy dissipated without causing reaction. This factor would account for the inner filtering effect. The extinction coefficient could be a variable if an absorbing species, other than that associated with the reaction, changes concentration. The attenuation factor is commonly supposed to be directly proportional to the molar concentration of the absorbing species. Then for species i pt =
The extensive literature on photokinetics shows that the overall rate of production of a product can depend upon the spectral distribution of the radiation source. Information on the characteristics of lamps (76A, 344 3B, 9B-I 7B, 76B) indicates that monochromatic light on an industrial scale is very difficult to produce. Also inserting monochromators between the lamp and reactor presents two difficulties : The intensity of the transmitted light is too low Severe limitations in reactor size and geometry are introduced, because the monochromator produces a directed beam concentrated in a small zone Further the monochromator is a solution only if the narrow wavelength band is needed for obtaining selectivity in the reaction. For bench scale research at least, gas, liquid, or glass filters may be useful for obtaining a narrow band of radiation. By using such filters (8A, 3B, 9B, IOB) the two listed difficulties may be overcome, but time instability and changes in transmission coefficient with temperature may cause problems. Also the question of low efficiency remains. Therefore, for engineering application polychromatic light will normally be used and a method is necessary to account for a range of wavelengths. Two aspects are involved: first, a method of evaluating the absorption of polychromatic light, and second, the effect of wavelength distribution on the rate, although they may be coupled. For the first problem the data necessary are the emission spectra of the lamp and the attenuation coefficients, which are functions of X (see Equation 20) for all the chemical species present. With this information there are two methods of approach. In the first the effects of polychromatic light on the absorption are treated by using an average attenuation coefficient calculated from the equation l ; P k I w ,xdX
where aiis the molar absorption coefficient. If there is absorption of radiation by several components in a system, and the concentrations are not unduly large, the assumption of additive absorption may be valid. If x1 is the mole fraction of species i, this approach gives for the total attenuation coefficient, p T PT
POLYCHROMATIC L I G H T SOURCES
The mass, energy, and radiation equations are coupled, so that an analytic solution, even for monochromatic light, is generally not possible. A special case where a solution is possible (45A) has been discussed. Note that this required linear kinetics and the assumption of a constant attenuation coefficient, despite Equation 20. For isothermal conditions it is sometimes possible to solve the mass and radiation equations by machine computation. Wall reactions are taken into account by proper boundary conditions.
where A 1 and A2 are the limits of absorption and/or the radiation of the lamp. If this value of 1.1 is to be employed later in estimating l a in the reaction rate expression, the assumption has already been made that all wavelengths of light are equally efficient for the primary chemical step, that is, the activation reaction associated with the light adsorption. To avoid this assumption and thus be able to account for the variation of kinetic constants with A, a second approach is to evaluate the fraction of the total radiation of the lamp associated with -each increment of wavelength Ah. Then Equation 19 is applied for each AA using the appropriate values of 1.1 and I,. This gives a set of I - X values which can be used directly in the rate of reaction term in the mass balance Equations 16. The decision as to which method to use depends VOL. 5 9
NO. 1 J A N U A R Y 1 9 6 7 ' 23
upon the reaction system, but the second more general approach requires more data and increases the computational work greatly, as noted in the subsequent section. Only the second method is correct if the intensity of the lamp is not constant at all wavelengths.
THE REACTION RATE The formulation of the equation for the rate of a photochemical process is one of the most difficult parts of the whole reactor design. I n many conventional reactors it is possible to write an empirical rate expression independent of transport processes. This is the concept of microkinetics (32A) where the reactor dimensions and level of transport processes are divorced from the kinetics. Such a procedure simplifies greatly the solution of the design equations. However, the concept loses its validity when the reaction cannot be represented by simple kinetics. This is the rule for photochemical systems. Only for exceptional instances, for example, actinometer reactions at restricted conditions, can the microkinetic model be confidently used. The development of the rate for photokinetics requires answers to the following problems : Effect of wavelength Reaction mechanisms Heterogeneous processes Diffusional effects Correction for dark reaction Each of these is discussed here and then the formulation of a rate equation is considered later. Effect of Wavelength
A rate equation which is independent of wavelength must satisfy two assumptions : The primary absorption step is independent of X All the secondary reactions follow the same paths over the range of wavelengths of the light The data available do not support either assumption as a general rule. Hence they can be accepted only after experimental verification. This problem constitutes a challenging opportunity for research and the fascinating possibility for lamp manufacturers and reactor designers working together to optimize a specific photoreaction. The Primary Step. A careful analysis of the initiation (primary, or activation) step is necessary to develop a sound rate equation. A detailed study is in the field of the spectroscopist and the aid of such a specialist is invaluable for complex photochemical systems. Nevertheless a simplified concept is worthwhile for the chemical engineer and a brief description will be given (see references 25A and 26A for more complete discussion). Important photochemical procerses may involve absorption of light by atoms and diatomic or polyatomic molecules. Absorption by atoms leads to electronic spectra, discrete up to the photoionization limit and continuous thereafter. The most important example is the absorption of short wavelengths by mercury. The 24
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
application is the catalysis of other reactions by activated mercury atoms-photosensitized reactions (734 78B). For example, methane absorbs radiation below 1400 A. directly producing CH2 CH4
However, if mercury atoms are used as a catalyst and subjected to radiation of 2537 A,, the following reactions occur : Hg ihv 4 Hg" (The asterisk denotes an activated species or free radical.) Hg"
+ H + Hg
The effect of the nature of the primary step and the wavelength on the reaction of CH4 is clearly evident. The absorption of light by an atomlike mercury is not a simple process. I t follows Lambert's law, but many variations can occur: chemical quenching such as shown above with methane, deactivation by collision, physical quenching (fluorescence and phosphorescence), dirnerization, reactivation, and activation to a higher level by absorption of more than one quantum of light, for example. Hence, the application of Lambert's law, without considering the possible use of energy in ways that do not cause reaction, could lead to erroneous results for atornic absorption. Absorption by diatomic molecules can lead to rotational, vibrational, or electronic spectra. The first two involve small quantified levels of energy, generally insufficient to produce photoreactions. The electronic energy levels are similar to those of atoms. Hence we again face the possibilities of many primary processes resulting from light absorption. Depending upon the type of molecule and the wavelength, absorption in the continuous region can cause either photoionization or dissociation. The latter is particularly important for photoreactions, either direct or of the photosensitized type. Different activated states are possible depending again upon wavelength, and subsequently the activated species may follow a multiplicity of paths. A careful analysis of the absorption spectra gives valuable information for any kinetic study, but the most important results for the reactor designer will be the amount of energy and the wavelength range of absorption. A good example is the study of the absorption of radiation by chlorine (22A). The analysis of absorption spectra of polyatomic molecules grows in difficulty with the complexity of their structure. The primary effects of light on a polyatomic molecule are physical (fluorescence and phosphorescence) and chemical (dissociation, photosensitization, isomerization, and dimerization). The chemical effects are of major importance for photoreactions. I t is clear from the diversity of the primary steps, more than one of which may be possible in a given case, that a very large number of secondary products are possible. Additional changes may occur when reaction mixtures can exist in more than one phase. In the vapor phase photolysis of cetones the primary step gives carbon monoxide and
two possible free radicals. When the cetone is dissolved in liquid isopropyl alcohol, photoreduction occurs giving pinacol. This problem has been studied extensively using benzophenone as a reactant (40A). I t is hoped that this greatly simplified view of the primary process will suggest that a detailed analysis of the absorption characteristics of the reactants and products is the first step of a quantitative and fundamental reactor design. An illustration of this viewpoint is the paper by Noyes and co-workers (394. After the primary step has occurred, distinct energy levels are associated with each possible activated state of the atom or molecule. Thus the products of the primary step are dependent upon the wavelength. Only when the specific primary step is well understood would it be valid to assume that the kinetic constant is independent of X. Even then it is necessary that the volumetric rate of light absorption leading to the desired product is properly accounted for. I n other words, if the rate is formulated as Qt
kt will be independent of X only if la is for the primary process that causes reaction. For example, for a chlorination the primary process would be dissociation of Clz. This concept is in agreement with the second fundamental law of photochemistry (Stark, Einstein, Bodestein) which states that the absorption is a singlequantum process so that the sum of the quantum yields for all the primary processes must be unity. Thus kl would be unity in the following summation Z
= Z: kt(l~),=
Unfortunately the preestimation of the conditions for
ka being independent of X is difficult. Also the experimental verification by spectroscopic study is a major effect. Another approach, suitable for the design problem is to study the kinetics of the desired overall reaction at different wavelengths. This procedure does not test whether kt is independent of X but rather determines the combined dependence of both primary and secondary steps upon wavelength. Reaction Path. The secondary reactions are thermal rather than photochemical. Nevertheless, the wavelength of absorption frequently affects their path. For example, the yield of C O in the photolysis of acetone changes notably with A. Acetone has a continuous absorption spectra from 2235-2950' A. and a second, strong absorption below 2200 A. (39.4). The main reaction products are COYethane, and biacetyl. Davis (744 reports that the primary step is the formation of an acetyl radical: CHgCOCHa
The overall activation energy for the dissociation of the acetyl radical (to CO) is estimated to be 10-18 kcal./ mole, while about 70 kcal./mole are required for breaking the C-C bond in acetone. Therefore, Davis concluded that the acetyl radical, after absorbing light at
3130 A. would have 13-14 kcal./mole of excess energy. This is in the range of that necessary to decompose the acetyl radical :
CHaCO" --t CH3
For absorption at shorter wavelengths, additional excess energy would be available. This would be expected to increase the formation of C O according to Equation 26. At higher X values CO production should be reduced. The relation between wavelength and energy is 2.858 X lo6 = kcal. /gram-mole X A.
If the reaction products change with wavelength a sound method of treating polychromatic light becomes extremely difficult. If only the rate constants change, the second procedure, described previously may be used, once the rate constant-X relationship is known. Reaction Mechanism
Two types of single photoreactions are possible : Those whose product is obtained in the primary process; for example, isomerizations, dimerizations, hydrogen abstraction Those where the same reaction is produced in a single step after chemical quenching with a light-activated atom or molecule. The oxalic acid decomposition, under certain conditions, is of this type However, the vast majority of photoreactions appear to follow a complex mechanism consisting basically of primary, propagation, and termination steps but with many variations possible (see 7 723). We can illustrate the situation, where the primary step is dissociation, by considering the overall chlorination R-H
The reaction sequence could include: Primary step Propagation Termination (homog en eous) Termination (heterogeneous)
Clz 2 c1* C1* R-H HC1 R* C12 C1* R-C1 R* R* R* R-R C1" c1* Clz C 1 e R-C1 R* R * + end products C1* end products
This description includes eight reactions, and with more products the situation is more complicated, as, for example, in the acetone reaction (74.4). Some simplification is obviously necessary, and here the results of the physical chemists' work may be of considerable help. Frequently, the choice of the limiting reactions is the key point of the whole analysis (8A). Heterogeneous Processes
Besides affecting directly the rate equation, reactions at the wall of photoreactors can lead to deposits which VOL. 5 9
affect the spectral distribution and intensity of the radiation reaching the reacting fluid. These factors are responsible for much of the difficulty in reproducing data in photochemical kinetics. If the wall reactions produce a deposit (84 IOA, 72A, 7 7 4 54A), no methods are available for treating the effects quantitatively. If the wall reaction achieves a stationary state without affecting the light transmission (IOA), the proper way to account for it is by use of proper diffusive fluxes and boundary conditions. If the same chemical species take part in homogeneous reactions, the microkinetics concept is not valid. Here the approach is to apply the wall reaction as a boundary condition applicable to the mass balance equation (Equation 16) for that species. Simplified forms of this problem, wall-catalyzed reaction with no homogeneous reaction, have been solved ( 3 I A , 538). Also Bateman (4A) has obtained solutions of differential equations of the type describing mass balances for the case of a firstorder homogeneous reaction combined with a firstorder wall reaction (linear boundary condition). Chain reactions with coupled sets of mass balance equations have not been considered. If the wall termination steps are treated as homogeneous reactions, the result is rate constants which are functions of the surface-to-volume ratio. The significance of wall reactions can be evaluated by varying this ratio or by changing the pressure for gaseous systems. This latter variable affects the difTusion rate and hence concentration at the wall. The evidence available ( 7 7B) suggests that wall reactions should be treated as first-order processes. It would be helpful to be able to predict the relative significance of wall and homogeneous reactions. The problem has been studied, for single reactions (79A, 278, 368, 38A, 468, 4 9 4 2 B ) , but the results are of little value for complex and light-activated systems. At present they must be included or discarded from the formulation of the overall rate on the basis of available chemical information, observations (for example, wall deposits), and intuition. Diffusional Effects
The second term in the mass balance equation (Equation 16) accounts for mass transfer by molecular and turbulent diffusion. Since neither data nor direct methods of measuring diffusivities of activated species are established, it is difficult to make more than an order of magnitude analysis for diffusion. This approach introduces uncertainty in the rate constants obtained for complex kinetics. Also, the validity of extrapolating data on one reactor to another of different size or geometry is questionable without taking diffusion into account. This is especially critical when heterogeneous termination steps, or light heterogeneity, are present. Tentative studies have been made (28A, 418-43A) but the problem has not been solved. Recently R. Noyes (334 37A) has proposed using light intermittency techniques to study the diffusion 26
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
problem. This may be a valuable experimental method. I t should be mentioned that application of the stationary-state hypothesis for obtaining the overall rate equation is not valid when diffusional effects are significant (in the direction of the light path). Correction for Dark Reaction
The possibility of the desired product being formed in the absence of radiation was originally discussed by Noyes and Leighton ( I I B ) . I t has been observed in a chlorination reaction (8A), where earlier suppositions (54A) were involved, that it did not take place at room temperature. In terms of separating the dark reaction from the photochemical one, three cases are of interest. First, suppose that the two reactions occur in parallel but that the extent of the dark reaction is relatively small. A good approximation for the photoprocess is achieved by measuring the conversion for the dark reaction in the same reactor and subtracting this from the total value measured for the lighted case. Second, for extensive dark reaction this procedure can lead to significant errors, if the two reaction paths have the same secondary steps. A simple analysis, supposing a completely mixed reactor and employing the stationary state hypothesis, shows that the subtraction procedure is valid only if the rate of the photoreaction is first order in both reactant concentration and light intensity, and the dark reaction is also first order. For other kinetics a valid correction for the dark reaction requires a knowledge of both dark and lighted reaction kinetics. The third situation applies to dark reaction in the equipment upstream from the photoreactor. Here the situation may be like either the first or second case. However, consideration must be given to possible effects of the product on the behavior in the photoreaction.
F O R M U L A T I O N OF T H E RATE EQUATION The methods for treating polychromatic light and reactions already have been discussed so the analysis from here on will be based upon monochromatic radiation and a single reaction path (but with many steps). First is given a reasonably rigorous approach, pointing out the difficulties in its application. By imposing restrictions we then proceed to simpler equations for which solutions are possible. Rigorous Formulation
To be specific, a reaction system involving a twocenter chain carrier with both homogeneous and heterogeneous terminations will be used. The reactor chosen is a steady-state, tubular-flow type with streamline flow exposed to radial radiation. Isothermal and isobaric behavior is assumed along with constant physical properties. Suppose the reaction, which could represent some hydrocarbon halogenations, occurs in the presence of an inert M. The overall reaction is
The sequence of steps is assumed to be as follows : A2
+ hv -% 2 A*
+ RH 5 HA + R* R* + A& RA + A*
+ M 2 A2 + M
An analytical solution of this system of equations is beyond expectation and a numerical approach may not be feasible because of the many kinetic constants, the difficulty in estimating values of diffusivities for activated species, and the economic limitations of computer time. Practical considerations thus force us to seek simplifications.
2 R* 5 R2
Simplification of the Equations
+ R* f RA wall product A* + W W + wall product R* -+ W
If a plug-flow model is assumed, all radial gradients disappear in the mass balance equations (Equation 27). Under these conditions the concentrations of all species would obey the stoichiometric relationships. Wall reaction rate constants would appear only as functions of the surface-volume ratio. Thus the differential mass balance equations would be simplified and reduced in number of stoichiometric relationships. Also the wall boundary conditions are simplified to the zero concentration gradient requirement for all the species. An example of this simplification will be presented. For photosensitized reactions the absorption coefficient is independent of the extent of the desired reaction. This decouples the mass balance and radiation equations and thus simplifies the solution for this special type of process. When the absorption is very weak, additional simplifications are possible.
All the reactions are supposed to be irreversible; A2 is the only species which absorbs radiation, and it undergoes only dissociation. Methods of treating dark reaction have been discussed so we only consider here the photochemical process. For uniformly radial and unidirectional light, neglecting axial diffusion and end effects, the conservation equations for laminar flow reduce to
An additional hypothesis, which further simplifies the plug-flow model, supposes that the concentration of all intermediates is low and independent of time in a batch reactor, or independent of position (dC,*/dz = 0) for a flow reactor. Inherent in this hypothesis is that the initial buildup of steady concentrations occurs very rapidly, even for fast reactions. For this simplification to be valid, reference to Equation 27 shows that the diffusion term must also be negligible for the intermediates. This can be true if either the diffusivities are very high (completely mixed reactor) or the lifetime of the species is so short that it disappears by reaction or deactivation before there is time for significant diffusion. Conditions of applicability of the hypothesis have been examined by Benson ( 5 4 6 A ) and Rice (&A), among others, for specific reaction sequences and reactor conditions.
ci(r,O) = Cto
I(R,z) = I, bCi_ (R,z) _ _-- 0 for species without wall reaction br
for species with wall reaction
The basic difficulty in the solution of Equations 2 7 and 28 arises because they are coupled nonlinear differential equations. While solutions for the general case seem impossible, methods are at hand for certain simplified forms. A few of these will be mentioned, but it should be noted that they are applicable only for completely mixed or plug-flow reactors where the diffusion terms in the mass balances are eliminated. Aris (2A) shows how to obtain solutions for linear equations. I n general, no analytical solution is available for coupled sets of nonhomogeneous differential VOL. 5 9
NO. 1 J A N U A R Y 1 9 6 7
equations unless they are linear. For photochemical reactions this means that the exponential term in the radiation equation must be linearized and also that the various reaction steps are first order. Specifically, Chien ( I 7A), extending the work of Moelwyn-Hughes (35A)) has given solutions for two-step consecutive reactions. Several simple (nonchain) photoreactions could be described with these solutions. More complete treatments of complex reaction solutions are given by Benson (2B) and Frost and Pearson (7B). Wei and Prater (52A) have analyzed systems of coupled, linear differential equations with constant coefficients. A specific advantage of their approach is the ease which which rate constants could be determined from experimental kinetics data. Vasil'ev (@A) has extended earlier work on the Euler method of integrating sets of linear differential equations for any number of active species involved in the propagation steps. The method appears to be more practical for obtaining conversions, once the rate constants are known, than for evaluating these constants from experimental rate measurements.
= '/Z[(cRH)O - CRH
- CRA - cR*]
C H A = (CRH)O- CRH CRA = 2
- [(CRH)O - CRH] -
With these three expressions the numer of independent mass balance equations is reduced to four and these are:
REACTOR CALCULATIONS The two approximations described previously, the plug-flow model and stationary state hypothesis, permit reasonably simple solutions of the design equations. Both methods are applied in this section to the chain reaction system with nonlinear kinetics described earlier. Both approximations eliminate radial diffusion, velocity profiles, and wall reactions as boundary conditions. Note that the plug-flow model does include all termination steps including wall reactions. Otherwise, most of the complexities typical of real photochemical processes are retained in the analysis. The general equations describing the behavior of a steady flow, tubular reactor can be written assuming constant pressure, fully developed flow, a dilute system (Cbc > CA,),isothermal conditions, a narrow wavelength band, and that the kinetics of the individual steps follow their stoichiometry. There are several differential mass balances (for species A2, RH, A*, R", R2, HA, and RA), radiation equations, and boundary conditions. The Plug-Flow-Stoichiometric
We need not write out all the enumerated equations since approximations will be used for their solution. For the plug-flow model the following stoichiometric relations hold :
A. E. Cassano is Fellow of the Consejo Nacional de Investigaciones Cientificas y Tecnicas, of Argentina, on leave from Facultad de Ingenieria Quimica, Santa Fe, Argentina. P. L. Silveston is a member of the faculty of the University of Waterloo, Waterloo, Ont., Canada. J . M . Smith is Chairman and Professor of Chemical Engineering, The University of California, Davis. The authors acknowledge financial assistance from the Federal Water Pollution Control Administration in carrying out the work reported in thispaper. AUTHORS
INDUSTRIAL AND ENGINEERING CHEMISTRY
where ?7 is the constant velocity for the plug-flow model and ?a is the mean light adsorption per unit volume. This is obtained by integrating Equation 28 and gives 2 ffIwCA2
With the proper boundary conditions, Equations 35 to 39 can be solved by numerical methods to give the concentration of Az, or its conversion, along the length of the reactor. Stationary-State Hypothesis
With this added restraint the equations for the plugflow model can be further simplified. The additional restrictions are
These equations give algebraic expressions for Ca* and CR* which can be used to obtain the expression for the rate of disappearance of Az in terms of known concentrations. T o obtain results that are not unduly complex make some assumptions about the relative importance of the termination steps. Two cases will be evaluated separately: Only wall termination of R* is significant. With this restriction the rate equation is d c a-, dz
4 klk3otIWR (c-4~)~ 8(kw)R*
Only homogeneous termination of R* is significant. Then the rate is
I n Equations 41 and 42 the additional assumption has been made that the absorption is weak-i.e., CM >> CA,). This permits simplification of the radiation equation and the evaluation of Tu. Equation 41 or 42 replaces the mass balance and radiation Equations 35 to 39 for the plug-flow model. They can be integrated directly to obtain the axial concentration (of A2) profile along the reactor and represent a great simplification (microkinetics concept). illustrative Results
For complex kinetics the relative values of the rate constants can affect greatly the composition of the reactor effluent. In photoprocesses this situation is combined with uncertainties in estimating individual kinetic constants (few data are available for guiding the estimates). While the constants were estimated as carefully as possible the results that follow should not be regarded as representative of any real reaction system. Further, the comparison between plug-flow and stationary-state results will also depend upon the magnitudes of the constants. For example, if the stationary-state hypothesis is valid for one set of constants it would not necessarily be suitable for another set. Calculations were made for the plug-flow model and the two cases for the stationary-state model using the constants given in Table I. These values were estimated from the sources noted in the table. The plug-flow calculations were carried out numerically on an IBM 7040 computer using an Euler integration routine. The maximum stable step size was 1.2 X lo-'. The upper curves in Figure 1 show the resultant axial concentration profiles for A2 for the plug-flow model and for the stationary-state hypothesis based upon a homogeneous termination step (Equation 42). An intensity I, = 1 x lo-* Einsteins/(sec.)(sq. cm.) at the inside wall of a 2-cm. i.d. reactor was used. The level of I , is probably about the maximum expected for mono-
TABLE I .
RATE CONSTANTS AND PARAMETERS
Value 0 . 5 gram-mole/einstein 4.73 x 101a cc. (gram-mole) (sec.) 1. I 6 X I O 0 cc./(gram-mole) (sec.) 4.5 x 1016 cm.B/(grammole)*(sec.) I 014 cc./(gram-mole)(sec.) 1014 cc./( gram-mole)(sec. ) 11.4 cm./sec. 10.4 cm./sec. 5 x 106 sq. cm./mole 8X crn.-l 4.09 X 10-6 gram-mole/cc. 4.09 X IO-' gram-mole/cc.
Estimated Calculated from (37A) Estimated from ( 2 B ) and ( 78B) Estimated from ( 2 B ) Estimated from 2B) Estimated from ( 2 B ) Estimated from ( 2 B ) Estimated from (2%) Calculated from ( 8 A ) Estimated as 10% of aC0 1 atm. and 25' C.
1% of CY
chromatic light using normal light sources. The flow rate was chosen so that the residence time was 1.3 sec. Note that for these conditions the exit conversion, 1 CA,/(CA,)O,was only about 4% at the end of the reactor. For these conditions the stationary-state hypothesis gives results in good agreement with the more elaborate solution for the plug-flow model. The values of the homogeneous rate constants k4 and ke, and heterogeneous constants ( k , ) ~ * and (k&*, are small enough to have little effect on the plug-flow model results. Thus, a calculation based upon taking all these constants equal to zero shows no change from the curve given in Figure 1. Hence only kg is important. This makes the comparison between plug-flow and stationary-state models consistent, since Equation 42 is based upon ks being the only termination constant involved. Concentrations of the activated species R" and A*, as calculated by the plug-flow model, are also shown. Constant values were obtained at reactor lengths above z/L = verifying the stationary-state hypothesis throughout the practical length of the reactor. For comparison the stationary-state results based upon Equation 41 are also indicated in Figure 1. As mentioned, the rate constant for this termination step, (k,)R*, is relatively much less than kg. As a result higher conversions are obtained. The large difference between the two curves for the stationary state hypothesis demonstrates the importance of choosing an adequate mechanism for the sequence of steps involved.
TEMPERATURE EFFECT In principle the effect of temperature on the overall rate of reaction will be determined by the activation energies of all the important steps and the effect of temperature on la. The latter arises from the variation of the absorptivity a with temperature. In practice the problem may be relatively simple. This is because overall activation energies are very low (see B references). Of course operating at different temperature levels can change the nature of the primary or secondary steps and cause large changes in the type and distribution of products. An example is the photolysis of acetone at wavelengths above 2600 A. At room temperature considerable diacetyl is produced. At higher temperatures diacetyl undergoes thermal decomposition so that a remarkable decrease in its production as a final product is observed (74A). Above 120' C. several investigators report no diacetyl in the product. I n the remainder of this section we are interested only in the effect of temperature changes with the same reaction path. The energy added to a molecule at a given wavelength by increasing the temperature is small compared to the energy accompanying the absorption of a quantum of light. Hence, if the proper absorptivity is used, the temperature should not have a significant effect on the primary process. An exception could arise if dissociation is not directly obtained by light absorption. Then the additional increment of energy provided by increasing the temperature might be sufficient to cause disVOL 59