Temperature Distribution around the Resting or Moving Solid-Liquid Interface in a Rod

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We have considered the temperature distribution around the solid-liquid interface in a rod as a function of the furnace temperature gradient, G_f, and of the moving velocity of the interface or velocity of crystallization, R_i, for the case of rod resting or moving at a constant speed, R, in an infinite long cylindrical vacuum furnace with a uniform temperature gradient. It is assumed that the solid-liquid interface is planer even if it is moving so that R_i is very small and that the temperature distribution in a normal cross-section of the rod is uniform so that the convection of heat does not occur in the liquid. Let us denote the temperature of the furnace as T_f, the cross-sectional area of the rod as A, its circumference as S, its density as ρ, its specific heat as c, its thermal conductivity as k, its emissivity as σ, its temperature as T, its temperature gradient as G, its melting point as T_E and its latent heat of solidification as L, the positional coordinate along the rod axis as x, and the time as t. We express quantities associated with the solid and with the liquid, respectively, by putting suffixes s and l. Further, let us take x=0 at the interface separating the solid portion (x<0) and the liquid one (x>0) . Then, we get a relation that the velocity of crystallization is the same in magnitude as the moving velocity of the rod. We also obtain, for small G_f and R_i, the following relations : -T_<fo>=TE-{(BG_f+DR_i)/2 T_E^<3/2>'} , G_<lo>=Gf-K_l^<1/2>(BG_f+DR_i), G_<so>= G_f+K_s^<1/2>(BG_f+DR_i), T_l=T_f+{(BG_f+DR_i)/2 T_E^<3/2>} cxp[-(R_i/2 a_l)-2K_l^<1/2>T_E^<3/2>{1-(3/8)(BG_f+DR_i)/T_E^<5/2>} x], and T_s=T_f+{(BG_f+DR_i)/2 T_E^lt;3/2>} exp [-(R_i/2a_l)+2K_s^<1/2>T_E^<3/2>{1-(3/8)(BG_f+DR_i)/T_E^<5/2>}x[ , where K=(S/A) (σ/k), α= k/ρ_c, B=(k_lA_l-k_sA_s)/(S_lA_lK_lσ_l)^<1/2>+(S_sA_sK_sσ_s) ^<1/2>] and D=ρ_sA_sL/{(S_lA_lk_lσ_l)^<1/2>+ (S_sA_sk_sσ_s)^<1/2>}. These equations indicate that, for the resting interface (R_i=0), T_E〓T_<fo> and G_<lo>〓G_f〓G_<so> accroding as k_lA_l〓k_sA_s for common metals or for bismuth, and that, for the moving interface during the proceeding of solidification (R_i>0), as R_i increases, eventually T_<fo> becomes lower than T_E, G_<lo> becomes negative, and G_<so> becomes greater than G_f, even in the case of k_lA_l<k_sA_s. These results of calculation are found to accord well with the results of Goss and Weintroub's experiments with a rod resting in a moving furnace, which measured (∂T/∂t)_x=R_iG.
東北大学の論文

著者

Yamamoto Mikio
The Research Institute For Iron Steel And Other Metals

Temperature Distribution around the Resting or Moving Solid-Liquid Interface in a Rod

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