Ind. Eng. Chem. Res. 1988,27,1087-1095
1087
KINETICS AND CATALYSIS Modeling of a Gas-Liquid Tank Photoreactor Irradiated from the Bottom. 1. Theory Orlando M. Alfanot and Albert0 E. Cassano*t INTEC,%Casilla de Correo No. 91, 3000 Santa Fe, Argentina
A semibatch stirred tank photoreactor irradiated from the bottom is studied. The radiation emitting system is a tubular source located at the focal axis of a parabolic reflector. A consecutive chain reaction-the gas-liquid chlorinations of trichloroethylene and pentachloroethane-is used for the application of the model. The model presents the mass balance equations in the liquid bulk considering two mixing states: perfectly mixed free radicals and nonmixed free radicals. A system of firsborder integrodifferential equations with initial values is obtained and then it is solved numerically. The differences in the predicted conversion between both models are analyzed as a function of the optical thickness of the medium. It is found that both models coincide only when the optical density is negligible and that, in general, the conversions obtained with free radicals in a state of perfect mixing are always higher than those predicted with free radicals in a nonmixing state. The design of a two-phase reactor activated by radiation requires the consideration of various aspects, many of them common to those found in the study of conventional reactors and others derived from the intrinsic characteristics of photoreactors. Among the former, we can include the difficulties originated in the analysis of a complex chemical reaction, the incorporation of mass transfer between phases, and the possible existence of different mixing states. The latter aspects, on the other hand, demand the solution of problems derived from the inevitable interaction between the radiant energy emitter (UV source) and the energy receptor (absorbing species) and from the intrinsic nonuniformity of the radiation field. The relationship “radiant energy emitter-radiant energy receptor” requires the study of the emitting system with adequate rigorousness;the modeling of the energy emission process, produced by a UV radiation source and a reflecting mirror (to improve the energy efficiency), leads to a complex development. Present-day simple models, frequently employed to describe the source emission process, fail to produce good results when the emitting system includes a curved reflecting mirror. Another problem connected with this interaction appears when the polychromatic nature of that interaction is considered. Commercially available radiation sources emit in different regions of the UV and visible range; the walls through which the radiation enters as well as the reactants also absorb thisenergy in a variable way with respect to the wavelength under consideration. The reflection coefficients deserve a similar comment. All this information must be incorporated to the model. ~~~~
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‘Research Assistant from CONICET and U.N.L. *Member of CONICET’s Research Staff and Professor at U.N.L. 8 Instituto de Desarrollo Tecnoltjgico para la Industria Qdmica. Universidad Nacional del Litoral (U.N.L.) and Consejo Nacional de InvestigacionesCientificas y Tknicas (COMCET), 3000 Santa Fe, Argentina.
The effect produced by the radiation field nonuniformity on the reaction rate and on the absorption rate of the gas in the liquid should be stressed. A nonuniform radiation field produces a nonuniform reaction rate, and this could cause the mass transfer between phases to be spatially dependent. Besides, in systems where stable species and free radicals or atomic species coexist, even if the former were assumed perfectly mixed, the existence of concentration gradients of the latter cannot be disregarded. This would lead us to discuss the presence of different mixing states. The attenuation of radiation, resulting from the optical thickness of the reacting medium, helps to enhance the nonuniformity of the radiation field. Therefore, according to the magnitude that this property may acquire, there may appear interactions of the type “attenuationextent of the reaction”, and “attenuation-mixing state”, which will play a fundamental role in the behavior of the photoreactor. Finally, the radiation field distortion provoked by the presence of heterogeneities in the reacting medium should be pointed out. This effect, added to those aspects previously mentioned, would lead to the formulation of a highly complex model. Therefore, at this moment we have preferred to look for a partially simpli.fied solution with which to tackle this problem using an ueffectivenabsorption coefficient for the radiation attenuation. Only a few research papers have considered all these aspects of reactor modeling simultaneously. Papers using simplified models to describe the radiation field are quite numerous in the literature. Among them, we can list those papers including the linear emission models (Harris and Dranoff, 1965; Jacob and Dranoff, 1968; Akehata and Shirai, 1972) and those including the two- and three-dimensional incidence models (Matsuura and Smith, 1910; Harada et al., 1971; Zolner and Williams, 1971). Generally speaking, they have been employed to study more limited aspects of the problem such as the evaluation of kinetic constants and reaction rate expressions or the analysis of the mixing effects, with the intention of working with
0888-5885f 88f 2621-~081$01.50f 0 0 1988 American Chemical Society
1088 Ind. Eng. Chem. Res., Vol. 27, No. 7 , 1988 AG1TATOR
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4 THERMOMETER
1 Figure 2. Comparative analysis between the mean lifetime of all reacting species and the characteristic mixing time.
atmospheric pressure and room temperature, are used as typical gas-liquid systems.
Figure 1. Schematic diagram of the gas-liquid stirred tank reactor and the radiation emitting system.
either analytical or simpler numerical solutions. Less numerous are the papers using rigorous source models, among which we could cite the extense source models with volumetric emission (Irazoqui et al., 1973) and with superficial emission (Stramigioli et al., 1975; Yokota et al., 1976). These models have been mainly employed in the solution of particular cases of the system, as, for example, the calculation of the radiation field in a nonabsorbing medium, or for the analysis of reactors where a photosensitized reaction takes place. Their application to the study of homogeneous systems with complex kinetics, as in the caae of multiple-step reactions and chain reactions, is considerably more recent (Romero et al., 1983; Clariii, 1984; De Bernardez, 1984). On the other hand, the literature available in the area of heterogeneous photoreactor modeling is relatively scarce. We could mention the papers by Akehata et al. (1976), Yokota et al. (1977,1981), Spadoni et al. (1978), and Otake et al. (1981,1983). At present, this area has become the object of intense research work due to its particular technological potentiality in water photolysis. The main purpose of this paper is to present a model for a stirred tank reactor where a chain, consecutive gasliquid reaction takes place. The emitting system is made up of a tubular radiation source located along the focal axis of a cylindrical reflector of parabolic cross section which allows the irradiation of the reactor from the bottom. The theoretical analysis comprises the evaluation of the radiation field by means of the Extense Source Model with Volumetric Emission (ESVE model), the inclusion of two possible different mixing states for the reacting species, the formulation corresponding to the gas absorption rate in the bulk of the liquid, and the consideration of a chain mechanism for the expression of the rate of reaction. The m a s balance equations in the bulk of the liquid constitute an initial value problem in a system of first-order integrodifferential equations which must be solved numerically. The experimental verification of the model will be described in a forthcoming publication (part 2 ) where the trichloroethylene and pentachloroethane chlorinations, at
Modeling Equations Figure 1represents a simplified scheme of the gas-liquid heterogeneous reactor and of the emitting system that is used to irradiate the stirred tank from the bottom. This arrangement allows the isolation of the radiation source from the reacting system and, at the same time, facilitates the fulfillment of various requirements apart from routine cleaning, such as good mixing, reasonable cooling capacity to remove the heat produced by the reaction, and availability of construction materials compatible with the presence of highly corrosive reactants and products. Further details on dimensions, employed construction materials, and reactor operation will be described in part 2.
Mixing States. In order to include the effect of mixing on the formulation of the photoreactor model, it is necessary to perform a comparative analysis between the mean lifetime of the species involved in the reaction and the characteristic mixing time. This study allows us to distinguish three theoretically possible mixing states. As explained below, only the first two ones will be considered here. According to Felder and Hill (19701, the following mixing states are considered: state 1, perfect mixing for stable species and free radicals; state 2, perfect mixing for stable species and no mixing for free radicals; state 3, no mixing for stable species and free radicals. Figure 2 presents a schematic diagram in which the species mean lifetimes and characteristic mixing times are compared. State 1 implies a stirring vigorous enough as to cause the mixing time to be shorter than the mean lifetime of the stable species and free radicals. Both types of species may then be considered in a perfect mixing state. State 2 is characterized by a mixing time that is shorter than the mean lifetimes of the stable species but longer than that of the free radicals. This implies perfect mixing for the former and no mixing for the latter. Free radicals become immobile for the time scale of the mixing process, and it can be said that they “are born, live, and die” in the same position. State 3 is represented by a characteristic mixing time longer than the mean lifetime of both types of species. All chemical species involved in the reaction stay in a no-mixing state, the comments pertaining to the previous case being applicable to all species. Under the operating conditions of the experimental device that was employed, the existence of the third mixing state is very unlikely to occur. This third mixing state would require quite a long mixing time, which would imply a very poor stirring of the reacting mixture. Hence, only state 1and state 2 will be considered in this work. Nonetheless, they should be understood as limiting cases for the actual
Ind. Eng. Chem. Res., Vol. 27, No. 7, 1988 1089 Table I. Dimensionless Mass Balance Equations in the Liquid Bulk initial ____
mixing model 1 2
SS FR SS
FR
PM PM PM NM
integrodifferential equations conditions dt(7)/dr N(7) - ( Q ( x , s ) ) t(0) = +o d$(r)/dr = - ( Q ( x , T ) ) +(O) = 0 d$(r)/dr = N ( I )- ( Q ( x , ~ ) t(0) ) = d+(x,r)/& = - Q ( x , ~ ) IL(X,O) = 0
+,
SS = stable species; PM = perfect mixing; FR = free radicals; NM = no mixing.
mixing state of the system. Very often, in real situations results should fall between them. Mass Balance Equations in the Liquid Bulk. To simplify the nomenclature, the chemical species of the system may be identified by assigning each species a generic subindex (i) corresponding to the number of chlorine atoms that are present, as follows: HCl(+J, C12($2), :2HCl3($3), C-2HCld$J, C2CL3($6), Cl*($A, C2HC14'($4),and C2C15*($5).Thus, it is possible to define different dimensionless arrays for the concentrations of the stable species ($) and the free radicals ($), for the reaction rfites of the stable species (0) and of the free radicals (Q),and for the gas absorption rate (N). The mass balances corresponding to mixing states 1and 2 may be written as indicated in Table I. It may be observed that the concentrations of the stable species are always a function of time for both models, while the concentrations of the free radicals are only a function of time for model 1and a function of position and time for model 2. The rate of reaction is always a function of space and time, while the gas absorption rate has been considered independent of position. This latter assumption will be analyzed and justified in the following section. In order to complete the mathematical description of the system, we must find the expressions for the gas absorption rate and the reaction rate.
Gas Absorption Rate Vector N represents the absorption (or desorption) rate of the reacting species in the bulk of the liquid. In the previous section, it was assumed that N is independent of position but, actually, it is known that in its more general expression it could be a function of space and time. If we consider that only chlorine is the species that absorbs (or desorbs) in the liquid in a significant way, the dimensionless expression for N ( T )becomes Ki[$Pt(7) - $2(7)1 Ni(7) = 0 (i # 2) (1) It should be noticed that the two concentrations constituting the driving force are only a function of time, which results from the characteristics of either of the two mixing states under consideration. The dominant regime in which the reaction takes places has to be analyzed to establish the functional dependence of the gas absorption rate and, in addition, obtain a proposal to evaluate Kl. Simultaneously, it will be necessary to establish a mass balance in the gas phase to find an expression for $?*. Reaction Regime. To analyze the reaction regime, we shall follow the ideas proposed by Astarita (1966, 1967) and Astarita et al. (1983). This method consists in establishing a mass balance in the film for the species that is absorbed (chlorine), defining two characteristic times for the process: t D and t,. The first one represents a mean lifetime of the surface elements of the liquid in contact with the gas. The second one, instead, is a measure of the N2(7) =
time that is required to produce an appreciable extent of the reaction. By comparing the orders of magnitude of these two characteristic times, it is possible to draw conclusions with respect to the dominant reaction regime. Due to the nonuniformity of the radiation field and, consequently, to the nonuniformity of the reaction rate, it will be necessary to redefine the reaction time to adapt it to the analysis of a photochemical reactor. In this case, a mass balance in the film would have the form
in which it has been assumed that the reaction rate only depends on the position in the bulk of the liquid and on the concentration of the species that is being absorbed. We have defined a minimum reaction time which is given by
This is a reaction time evaluated at a position in which the radiation field (and consequently the reaction rate) is maximum. For the cylindrical geometry of the photoreactor irradiated from the bottom, position , x n, coincides with the center point of the reactor base. Thus, xmmn,
= ( r= ~ 0,YI = 01
(4)
In order to estimate the order of magnitude of tr,min,a global kinetic expression (Huybrechts et al., 1962; Chiltz et al., 1963) will be used for the chlorination of trichloroethylene as well as the value of ea estimated from the results of computing the radiation field at the position given by eq 4. Since it is a substitution reaction, a consideration of the chlorination of pentachloroethane is not necessary because it is known that it has to be considerably slower than that of chlorine addition to trichloroethylene; consequently, it would yield higher values of tr,min. Substituting all these results in eq 3, one obtains t r , m h z 10 8 (5) To evaluate t D , both the diffusion coefficient of chlorine in carbon tetrachloride and the mass-transfer coefficient for pure physical absorption should be estimated. The diffusion coefficient may be obtained from the measurements of Clegg and Tehrani (1973) for the system mentioned above, and the mass-transfer coefficient may be estimated from the correlation obtained by Calderbank and Moo Young (1961). Substituting the calculated values of both coefficients in the definition of t D , it yields A t D = Dim/(k1°)2 s (6) By comparing the reaction time and the diffusion time that were obtained, it can be observed that t~
