344
Ind. Eng. Chem. Fundam. 1980, 79, 344-351
It is difficult at the present stage of our study to determine to which step of the two possibilities the measured activation energy, i.e., 104 kJ mol-l, corresponds. The two possibilities make it difficult to discriminate the difference in the rate-limiting step between reactions of the allyl radical with acetylene and ethylene. However, the reported kinetic analyses are expected to contribute to the establishment of the elementary reactions related to the addition of allyl radical to unsaturated hydrocarbons. As for the addition step, the minimum declaration allowed in the present study is that the activation energy for the addition of allyl radical to ethylene or to acetylene should not exceed 48 kJ mol-’ which was obtained for cvcloDentene formation by Sakai et al. (1976), provided thatthe addition of the allyl radical to ethylene does not differ much from that of the allyl radical to acetylene.
Literature Cited Back, M, H,, Can, J . ‘-hem,, 49, 2199 (1971). Golden, D. M., Gac, N. A., Benson, S. W., J . Am. Chem. SOC.,91, 2136 (1969). Hougen, D, A,, Watson, K. M,, Process Principles,,, 884, Wiley, New York, 1943. Kerr, J. A., Parsonage, M. J., “Evaluated Kinetic Data on Gas Phase Addition Reactions, Reactions of Atoms and Radicals with Alkenes, Alkynes and Aromatic Compounds”, Butterworths, London, 1972. Nohara, D., Sakai, T.. Ind. Eng. Chem. Prod. Res. Dev., 12, 322 (1973). Nohara, D., Sakai, T., J . Jpn. Pet. Inst., 23, 133 (1980). Sakai, T., Nohara, D., Bull. Jpn. Pet. Inst.. 17, 212 (1975). Sakai, T., Soma, K., Sasaki, Y., Tominaga, H., Kunugi, T., A&. Chem. Ser., No. 97, 68 (1970). Sakai, T.. Nohara, D., Kunugi, T., ACS Symp. Ser., 32, 152 (1976). Vinokurov, D. M., Zabedenii, M. B., ~ z v vyssh. . uchebn. z a v d . m i m . Khim. Tekhnol., 6, 83 (1963).
Received for review July 18, 1979 Accepted M a y 27, 1980
Diffusion and Reaction in a Stagnant Boundary Layer about a Carbon Particle. 5. Pseudo-Steady-State Structure and Parameter Sensitivity S. Sundaresan and Neal R. Amundson’ University of Houston, Houston, Texas 77004
Models of different degrees of complexity describing diffusion and reaction in a stagnant boundary layer surrounding a carbon particle are constructed. The models include the heterogeneous combustion reactions with the intraparticle effects lumped at the particle surface in concert WWI radiation resulting in nonlinear boundary conditions. The genesis of steady-state structures associated with carbon combustion is identified. The sensitivity of the steady-state structures to the complexity of the model for diffusion and reaction in the boundary layer and to the choice of values for some key parameters appearing in the model is investigated. The qualitative features of the steady-state structures are sensitive only to the environment with which the particle is in radiant interaction and the O2content in the gas phase.
Introduction
The combustion of a carbon particle, due to its practical importance, has attracted an enormous amount of experimental as well as theoretical modeling efforts and the experimental evidence leads to the following picture. The combustion of a carbon particle involves diffusion of oxygen through a stagnant film and its reaction at the carbon surface to produce mainly CO at temperatures above 1273 K. As the CO diffuses away it is partially oxidized to COz. In addition, the resulting COScan be reduced to CO at the carbon surface. Significant CO oxidation in the boundary layer can lead to maxima in the temperature and the COS concentration profiles in the boundary layer as well as complete consumption of O2 in the boundary layer itself (Wicke and Wurzbacher, 1962; DeGraaf, 1965; Kish, 1967). The existing literature has been summarized in a recent paper by Caram and Amundson (1977). It is obvious that combustion of carbon is a transient process; however, before studying the dynamics of carbon combustion, it is useful to analyze the pseudo-steady-state (PSS) problem. Studying the PSS problem reveals the complete structure underlying the dynamics and it is indeed possible to predict the dynamics, at least qualitatively, from the results of such an analysis. The PSS problem has been studied rather extensively by a number of researchers, but most of the theoretical models de-
scribing the PSS problem may be classified according to their assumption as to where the oxidation of CO takes place, e.g., single film and double film models. The inherent weakness of these models is that their range of validity is not clearly defined. To resolve this difficulty, Caram and Amundson (1977) undertook a rigorous analysis of the equations of conservation of mass and energy in the boundary layer, while lumping all the intraparticle effects at the surface and explained the pathology in the system. They used a simplified model which employs Fick’s law of molecular diffusion and Fourier’s law of heat conduction together with the assumptions of independent diffusion with equal diffusivities for all the species, radiation equilibrium, and temperature-independent transport properties. Mon and Amundson (1978) extended the results of Caram and Amundson (1977) by taking into account the differences in diffusivity between the gaseous components while also considering radiant interaction between the particle surface and the environment. The Stefan-Maxwell equation, simplified by assuming negligible Stefan flow and small values of the mole fractions of gaseous reactants, were used to relate the fluxes to the composition gradients. Their results supported the experimental observation that negligible CO oxidation takes place in the boundary layer surrounding small particles (
