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Notes On Ivantsov's Solution: 1 Thermal Field Around A Dendrite Tip

1. The document summarizes Ivantsov's solution for the thermal field around a dendrite tip during solidification. It presents the governing equations and boundary conditions in dimensional and non-dimensional forms. 2. The equations are written for a paraboloidal dendrite tip shape in cylindrical coordinates moving at a constant velocity. Non-dimensional numbers like the Peclet number and Stefan number are defined. 3. The equations are transformed to a paraboloidal coordinate system where the solid-liquid interface is a simple boundary. Closed-form solutions for the temperature field are presented in 2D and 3D.

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Harris Daniel
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67 views2 pages

Notes On Ivantsov's Solution: 1 Thermal Field Around A Dendrite Tip

1. The document summarizes Ivantsov's solution for the thermal field around a dendrite tip during solidification. It presents the governing equations and boundary conditions in dimensional and non-dimensional forms. 2. The equations are written for a paraboloidal dendrite tip shape in cylindrical coordinates moving at a constant velocity. Non-dimensional numbers like the Peclet number and Stefan number are defined. 3. The equations are transformed to a paraboloidal coordinate system where the solid-liquid interface is a simple boundary. Closed-form solutions for the temperature field are presented in 2D and 3D.

Uploaded by

Harris Daniel
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Notes on Ivantsov’s solution

G. Phanikumar
July 18, 2011

1 Thermal field around a dendrite tip


1.1 Assumptions
• Dendrite grows at steady state velocity v
• Surface tension effects are negligible
• Dendrite grows in a shape perserving manner
• Changes of density upon solidification are negligible

1.2 Paraboloid case


Write the governing equations and boundary conditions in a coordinate system (z, r) fixed at the center of a
circle that fits the tip of the paraboloid and moves at a constant velocity v in the z direction. Let R0 be the
radius of the tip.
Shape of the parabola is given by a single parameter R0 as follows:

R0 r2
z= − = P (r; R0 ) (1)
2 2R0
The governing equation for the solid domain defined by : z ≤ P (r; R0 ) is
∂Ts
= αs ∇2 Ts
−v (2)
∂z
The governing equation for the liquid domain defined by : z ≥ P (r; R0 ) is
∂Tl
−v = αl ∇2 Tl (3)
∂z
Boundary condition at the interface at z = P (r; R0 ) is

Ts = Tl = Tm (4)

Stefan’s condition at the interface at z = P (r; R0 ) is

ks ∇Ts · n̂ − kl ∇Tl · n̂ = ρLf v n̂ (5)

Far field temperature at r2 + z 2 → ∞ is given by:

Tl = T∞ (6)

1.3 Non-dimensionalization
Scale the distances using R0 and the temperature using Tm − T∞ to bring dimensionless numbers out of the
equations above. Assume that the dendrite is isothermal such at ∇Ts = 0.
z
z∗ = (7)
R0
r
r∗ = (8)
R0
Tl − T∞
θl = (9)
Tm − T∞

1
We can rewrite the equations from the previous section as follows:
Shape of the parabola:
1 
z∗ = 1 − r∗ 2 = P ∗ (r∗ ) (10)
2
With hindsight, define the following two dimensionless numbers:
vR0
Pe ≡ (11)
2αl
Cpl (Tm − T∞ )
Ste ≡ (12)
Lf
Governing equation in the liquid domain where z ∗ < P ∗ (r∗ ) is

∂θl
−2Pe = ∇ 2 θl (13)
∂z ∗
Interface temperature at z ∗ = P ∗ (r∗ ) is
θl = 1 (14)
Stefan condition at z ∗ = P ∗ (r∗ ) is
2Pe
−∇θl · n̂ = n̂ (15)
Ste

1.4 Paraboloidal coordinate system


Define a paraboloidal coordinate system with ξ, η as below.

r∗ = ξη (16)

1 2
z∗ = ξ − η2

(17)
2
p
ξ 2 = z ∗ + r∗ 2 + z ∗ 2 (18)
p
η 2 = −z ∗ + r∗ 2 + z ∗ 2 (19)
In this coordinate system, the solid-liquid interface is at ξ = 1
The vector normal to the solid-liquid interface will then be ξ.ˆ
Thus the expression in Stefan’s condition can be simplified using:
∂θ
∇θl · n̂ = (20)
∂ξ

The solution (in 2D) is given by:


√  √ 
πPe
θl (ξ) = · exp (Pe) · erfc ξ Pe (21)
Ste
The solution (in 3D) is given by:

E1 Pe ξ 2
θl = (22)
E1 (Pe)
Where, the exponential integral is defined as
∞
e−s
Z
E1 (x) = ds (23)
x s

∂ e−x dx
E1 (x) = − (24)
∂ξ x dξ

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