In a case of an infinitely long cylindrical rod placed in an infinitely long cylindrical vacuum furnace of uniform temperature gradient, the temperature distribution in the rod near its solid-liquid interface where the crystalization is in progress is dealt with as a function of the furnace temperature gradient and onward velocity of the interface along the rod, or in other words, the velocity of crystal growth, when the rod is left at rest or moved lengthwise at a slow and uniform rate. Calculations are made on the assumption that the interface is planar and that the temperature in lateral cross-section of the rod is uniform, hence, no convection motion occurs in liquid state. The following notations are used in the calculations:
Gƒ (>0), furnace temperature gradient;
G, rod temperature gradient;
Ri, onward velocity of the solid-liquid interface along the rod;
R, velocity at which the rod is moved lengthwise;
Tƒ, temperature of the furnace;
T, temperature of rod;
A, its .cross-sectional area;
S, its circumference; ρ, density of rod material;
c, its specific heat;
k, its thermal conductivity; σ, its emissivity;
L, its latent heat of solidification;
t, time. Suffixes
s and
l signify solid and liquid states, respectively; the rod axis is taken as
x coordinate axis with the origin at the solid-liquid interface, solid side being positive, liquid side negative.
An important relation
Ri=-
R is obtained showing that the onward velocity of the interface by crystalization is the same in magnitude as the moving velocity of the rod. For small
Gƒ and |
Ri|,
Tƒ0=
TE-{(
BGƒ+
DRi)/2
TE3/2},
Gl0=
Gƒ-
Kl1/2(
BGƒ+
DRi),
Gs0=
Gƒ+
KS1/2(
BGƒ+
DRi), and
Tl=
Tƒ+{(
BGƒ+
DRi)/2
TE3/2} exp [-(
Ri/2α
l)-2
Kl1/2TE3/2{1-(3/8)(
BGƒ+
DRi)/
TE3/2}]
x,
Ts=
Tƒ+{(
BGƒ+
DRi)/2
TE3/2}exp [-(
Ri(2α
s)+2
Ks1/2TE3/2{1-(3/8)(
BGƒ+
DRi)]/
TE3/2}
x, where
K=(
S/
A)(σ/
k), α=
k/ρ
c,
B=(
klAl-
ksAS)/{(
SlAlklσ
l)
1/2+(
SsAsksρ
s)
1/2} and
D=σ
sAs
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