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We perform numerical simulations to study effects of alternating current (AC) frequency on thermal stress evolution during the Joule heating process in a 48-rod Siemens reactor. The characteristics of temperature, current density, and stress distributions within the rods located in different rings are analyzed first. A Joule heating model using AC is then proposed for flexible power adjustment. The voltage-current curves are obtained from analysis of rod radius, location, and skin depth, as well as the properties of the polysilicon. The results indicate that, during the Joule heating process, large stresses typically occur in the central regions of the silicon rods. The highest stress occurs at the center of the silicon rods located in the outer ring when a standard power frequency of 50 Hz is utilized. The results further indicate that von Mises stress significantly decreases as AC frequency increases. Additionally, we propose a novel Joule heating method that is useful for producing larger high-purity polycrystalline silicon rods by considering both low frequency and high frequency power. Based on an analysis of energy consumption during the Joule heating process, we recommend optimized power supplies for the 48-rod Siemens reactor.

In the present work, a Joule heating model by direct current (DC) has been developed for the rods arranged in a 48-rod Siemens reactor (SMS-48). The combined effect of heat dissipation (radiation, convection and reaction energy) and heat conduction produced by Joule effect has been examined by application of the developed model, taking into account variations of the rod radius and wall emissivity. The influence of the rod location, rod radius and wall emissivity on temperature profile and voltage-current curves during the electric heating process have been studied. Furthermore, the thermal and electrical behavior of silicon rod arranged in the SMS-48 and SMS-24 has been compared with each other. The dimensionless parameter ( p ) has also been introduced to investigate the effect of the number of rods and their arrangement on energy consumption. The interesting results show that the temperature profile of the rod arranged in the inner ring within SMS-48 is flatter than that in the SMS-24. The curve of current increases and becomes lower for the inner ring and keeps a linear relation for the outer ring in a manufacturing process. The required current and voltage increase when the wall emissivity increases. Enlarging the reactor capacity and meanwhile decreasing the dimensionless parameter ( p ) is an effective method to decrease energy consumption.

A parametric study of the flow field and mixing characteristics of a TiCl4 jet injected into an O2 crossflow in an oxidation reactor for titanium pigment production is numerically investigated. A three-dimensional computational fluid dynamics (CFD) simulation of the turbulent gas mixing model is performed. The effect of geometrical (reactor diameter, jet nozzle number n, jet nozzle diameter, jet nozzle spacing S) and flow parameters (momentum flux ratio J) on the penetration depth (h/R) and mixing quality of the gases is examined. The results are validated with available experimental data and a good agreement is obtained. We show that three stages: under-, optimum, and over-penetration, occur sequentially in the oxidation reactor with increasing J. The kidney-shaped structure, characteristic of a jet-in-crossflow is formed, which is blurred when the mixture of TiCl4 and O2 moves into the downstream fluid. The h/R value at a minimum temperature difference is 0.683 and 0.604 for n = 32 and 16, respectively, which are within industrial production data range of 0.56–0.72. The optimum range of S is between 3.25 and 7.35. h/R strongly depends on the only dimensionless parameter J/n2 expressed in terms of the geometrical and flow parameters via the relation: h/R = 0.7274 + 0.20228 ln (J/n2+0.04587). The TiCl4 concentration profile changes from a quasi-sine to quasi-cosine distribution with increasing J/n2. Both the mixing non-uniformity and the time to attain the optimum mixing quality of TiCl4 and O2 decrease first and then increase with increasing J/n2.

The vector diffraction of linearly polarized plane wave passing through a thin circular plate with a sub-wavelength radius is investigated by the finite element method. The transverse distribution and propagating evolution of the diffracted field with different intensity expressions are analyzed and compared. It shows that there are not only the component of the incident polarization, but also those of other two vertical directions in the near-field diffracted region. Regardless of the relationship between the wavelength and the radius of the circular plate, the diffracted field could evolve into a steady annular diffracted pattern, in which there is a central bright spot (so-called Poisson spot). There is an obvious difference of the diffracted properties between two intensity expressions from the square of electric field and the average magnitude of the Poynting vector as the propagation distance is very small, while they can be considered same with a large propagation distance (e.g. more than a wavelength).