We consider a nonstationary problem of determination of temperature in a circular concentric cylindrical channel of infinite length with thermally insulated surfaces filled with a moving medium under the influence of inductive heating of a conductor in the form of a cylindrical helical line (helix) synchronously rotating about the axis of symmetry of the channel. Problems of this kind are encountered in the process of thermal treatment of biomaterials with the use of the pyrolysis reaction when the helical surface of an auger moving a material mixture is inductively heated by electromagnetic fields. We solve these problem by using the Laplace integral transform with respect to time, Fourier integral transform with respect to the axial coordinate, and Fourier–Bessel expansions with respect to the angular and radial coordinates. It is shown that the temperature in the channel is, as a rule, described by a linearly increasing function of time. Weak oscillations caused by the rotation of the heated helix are imposed on this component. Low-amplitude oscillations of temperature are described by double Fourier–Bessel series whose coefficients are determined via the roots of a transcendental equation containing cross products of the derivatives of Bessel and Neumann functions. These roots are numerically determined by the “regula falsi” method. The numerical analysis reveals a substantial influence of the radius, pitch, and the angular rotational velocity of the helix, and the linear velocity of the medium in the channel on the formation of space and time characteristics of the microstructure of temperature field. In particular, we establish the conditions leading to the resonant amplification of the amplitude of quasimonochromatic oscillations of temperature in the channel.
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Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 64, No. 1, pp. 107–123, January–February, 2021.
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Piddubniak, O.P., Piddubniak, N.G. Nonstationary Temperature Distribution in a Thermally Insulated Concentric Cylindrical Channel with Biomass Moving Under the Influence of Rotation of an Electrically Heated Helix. J Math Sci 274, 708–729 (2023). https://doi.org/10.1007/s10958-023-06631-7
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DOI: https://doi.org/10.1007/s10958-023-06631-7