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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.

The Maxwell’s field equations and Ohm’s Law were added to energy equation to account for the electromagnetic contributions to heat transfer phenomena within silicon rod arranged in an industrial Siemens reactor. Based on the combined effect of the heat dissipation (radiation, convection and reaction energy), a Joule heating model using alternating current (AC) has been developed. The analysis accounts for the rod radius, location, skin depth, as well as the properties of the polysilicon. The results indicate that high frequency current sources can generate an even temperature profile, and a higher deposition radius can be obtained. Alternating current, having a fixed or variable high frequency in the range of about 2.4 kHz to 500 kHz, is provided to concentrate at least 70% of the current in an annular region that is the outer 15% of a growing rod due to the skin effect. The voltage-current curves have been predicted based on the present model. Finally, a novel Joule heating method useful for producing the larger high-purity polycrystalline silicon rods by considering the low frequency power and the high frequency power is proposed. It is interesting that the average energy consumption ( AE ) can be decreased from 55 kWh/kg to about 44 kWh/kg as extending the maximum deposition radius from 7 cm to 10 cm by using a high-frequency current source.

This paper presents an electrical heating model using alternating current (AC) for the silicon rods located in a 24-rod Siemens reactor. In this model, the combined effects of heat dissipation (radiation, convection, and reaction energy), skin effect, and heat conduction owing to Joule effect are examined. The presence of the skin effect, which yields an important radial temperature profile controlled by the heat conduction equation for the rods, has been considered. The present model is validated using industrial current data, and it is observed that the numerical results are in good agreement with them. The influence of the location of the silicon rods, AC frequency, radius of rod, and wall emissivity on the temperature profile and current density has been studied through the application of the developed model. Tailoring the temperature profile of silicon rods and extending the maximum deposition radius has also been performed. Voltage-current curves, which are applied to generate the required heat during the manufacturing period, have also been studied using different AC frequencies and wall emissivities.

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).