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A 1-D model for highly sensitive tubular reactors

Description: We consider the steady state operation of wall-cooled, fixed-bed tubular reactors. In these reactors the temperature rise ..delta..T must normally be limited to small fractions of the adiabatic temperature rise ..delta..T/sub ad/, both to avoid runaway and maintain product selectivity. Yet ..delta..T/..delta..T/sub ad/ << 1 can only occur if eta = t/sub dif//t/sub reac/ << 1, where t/sub dif/ is the timescale on which heat escapes the reactor by ''diffusing'' to the cooled walls, and t/sub reac/ is the timescale over which the reaction occurs. So here we use asymptotic methods based on eta << 1 to analyze the 2-d reactor equations, and find the radial concentration and temperature profiles to leading order in eta. We then obtain a 1-d model of the reactor by substituting these asymptotically correct profiles into the reactor equations and averaging over r. This model, the ..cap alpha..-model, is identical to the standard (Beek and Singer) 1-d model, except that the reactor's overall heat transfer coefficient U is a decreasing function of the temperature rise ..delta..T. This occurs because as ..delta..T increases, the reaction becomes increasingly concentrated near r = 0, causing a decreased heat transfer efficiency through the reactor's walls. By comparing it with numerical solutions of the original 2-d reactor equations, we find that the ..cap alpha..-model simulates the 2-d equations very accurately, even for highly sensitive reactors operated near runaway. We also find that a runaway criterion derived from the ..cap alpha..-model predicts the runaway transition of the original 2-d equations accurately, especially for highly sensitive reactors. 19 refs.
Date: January 1, 1987
Creator: Hagan, P.S.; Herskowitz, M. & Pirkle, J.C.
Partner: UNT Libraries Government Documents Department

Derivation of the high field semiconductor equations

Description: Electron and hole densities evolve in x-z phase space according to Boltzmann equations. When the mean free path of the particles is short and electric force on the particles is weak, a well-known expansion can be used to solve the Boltzmann equation. This asymptotic solution shows that the spatial density of electrons and holes evolves according to diffusion-drift equations. As devices become smaller, electric fields become stronger, which renders the Basic Semiconductor Equations increasingly inaccurate. To remedy this problem, we use singular perturbation techniques to obtain a new asymptotic expansion for the Boltzmann equation. Like the Hilbert expansion, the new expansion requires the mean free path to be short compared to all macroscopic length scales. However, it does not require the electric forces to be weak. The new expansion shows that spatial densities obey diffusion-drift equations as before, but the diffusivity D and mobility {mu} turn out to be nonlinear functions of the electric field. In particular, our analysis determines the field-dependent mobilities {mu}(E) and diffusivities D(E) directly from the scattering operator. By carrying out this asymptotic expansion to higher order, we obtain the high frequency corrections to the drift velocity and diffusivity, and also the corrections due to gradients in the electric field. Remarkably, we find that Einsteins's relation is still satisfied, even with these corrections. The new diffusion-drift equations, together with Poissons' equation for the electric field, form the high-field semiconductor equations, which can be expected to be accurate regardless of the strength of the electric fields within the semiconductor. In addition, our analysis determines the entire momentum distribution of the particles, so we derive a very accurate first moment model for semi-conductors by substituting the asymptotically-correct distribution back into the Boltzmann equation and taking moments.
Date: January 1, 1991
Creator: Hagan, P. S.; Cox, R. W. & Wagner, B. A.
Partner: UNT Libraries Government Documents Department

Theory and simulations of Zone II microstructures in thin films

Description: The nature of the microstructure of vapor-deposited films is known to depend sensitively on the substrate temperature during deposition. At intermediate temperatures (T approx. .45 Tm where Tm is the melting point of the film) the film is made up of columnar-grains separated by metallurgical grain boundaries. Both an analytical and numerical analysis is presented in which the space-time evolution of the columnar microstructure (Zone II) is studied.
Date: January 1, 1987
Creator: Srolovitz, D.J.; Mazor, A.; Bukiet, B.G. & Hagan, P.S.
Partner: UNT Libraries Government Documents Department

Thin film microstructures: Simulation and theory

Description: The nature of the microstructure of physical vapor-deposited films depends sensitively on the substrate temperature during deposition. At low temperatures the microstructure is porous and ballistic aggregation-like, at intermediate temperatures the microstructure is columnar, and at elevated temperatures the grains are three dimensional. These different microstructural regimes are known as Zone I, II, and III, respectively. A theoretical analysis is presented in which the temporal evolution of the columnar microstructure (Zone II) is studied. The columnar microstructure is shown to be a balance between shadowing (which results in Zone I microstructures) and surface diffusion (which tends to smooth the surface). In addition to predicting the proper microstructure, this analysis properly predicts the temperature at which the Zone II to Zone I microstructural transition occurs. Since bulk diffusion is negligible and surface diffusion controls the microstructure in Zone II, the microstructure in the bulk of the film, may be viewed as frozen and all microstructural evolution occurs at the current, or active, surface. A Monte Carlo computer simulation technique which models the microstructural evolution of the surface is presented. The simulation follows the temporal evolution of realistic three dimensional Zone II microstructures and accounts for growth competition between adjacent grains and the formation of film texture.
Date: August 1, 1987
Creator: Mazor, A.; Srolovitz, D.J.; Hagan, P.S. & Bukiet, B.G.
Partner: UNT Libraries Government Documents Department