The influence of distribution intensity form of heat flux on the temperature by the local quasi-stationary heating on external surface of the ring

Abstract

The solution of the mixed quasi-stationary heat conduction problem for the ring has been obtained by using a finite Fourier transformation. The heating on the external side surface of the ring due to any locally distributed heat flux, as well as the convective cooling from internal side surface and from the end faces are considered. It has been proved that for some special geometrical and thermo-physical parameters of problem the received solution agrees with well known studies. The influence of some distribution intensity form of heat flux on the maximum temperature was studied. List of symbolsBi = h R/K dimensionless Biot number associated with the external side surface of the ring Bi 0 = h 0 R/K dimensionless Biot number associated with the internal side surface of the ring Bi = h R/K dimensionless Biot number associated with the end faces of the ring h heat transfer coefficient on external side surface of the ring (W m −2 K −1 ) h 0 heat transfer coefficient on the internal side surface of the ring (W m −2 K −1 ) h heat transfer coefficient on the end faces of the ring (modified Bessel function of the first kind of the order k(k = 1, 2) k thermal diffusivity of the ring (m 2 s −1 ) K thermal conductivity of the ringmodified Bessel function of the second kind of the order k(k = 1, 2) n non-negative integer Pe = ω R 2 /k dimensionless Peclet number q 0 characteristic (maximum) intensity of the heat flux (W m − 2) q * (θ ) dimensionless function, describing the change of intensity of the heat flux in circumferential direction Q total rate of friction heat directed into the ring (W) r radial coordinate (m) R external radius (m) R 0 internal radius (m) T temperature rise (K) T V bulk temperature of the ring (K) 768 A. Yevtushenko, J. Tolstoj-Sienkiewicz T f flash temperature on the external surface of the ring (K)flash temperature on the external surface of the ring T * * f = T * f /(2θ 0 ) dimensionless flash temperature on the external surface of the ringGreek symbols δ thickness of the ring (m) = δ/R dimensionless thickness of the ring θ circumferential coordinate (rad) θ 0 half of the contact angle (rad) θ * = θ/2θ 0 dimensionless circumferential coordinate ρ = r/R dimensionless radial coordinate ρ 0 = R 0 /R dimensionless internal radius σ = Bi / dimensionless parameter ω rotational speed of the ring (rad s −1 )