A hypothetical binary eutectic phase diagram of A – B is shown below. An alloy with 5 wt.% B solidifies with no convection. Assuming steady state, the critical temperature gradient (in K $mm^{-1}$) required to maintain planar solidification front is: ________ (round off to nearest integer). Given:
Diffusivity of B in liquid = $10^{-9}$ $m^2$ $s^{-1}$
Velocity of solidification front = 4 $\mu m$ $s^{-1}$
To determine the critical temperature gradient required to maintain a planar solidification front, we apply the equation for the critical temperature gradient \( G_c \) given by:
\( G_c = \frac{mL \cdot C_s \cdot \Delta C}{D \cdot V} \)
Where:
- \( mL \) = Liquidus slope
- \( C_s \) = Initial concentration of solute in solid (approx. solute in liquid for small concentrations)
- \( \Delta C \) = Change in concentration (approx. \( C_l - C_s \))
- \( D \) = Diffusivity of solute in liquid = \( 10^{-9} \, m^2/s \)
- \( V \) = Velocity of solidification front = \( 4 \times 10^{-6} \, m/s \)
Step 1: Determine the Liquidus Slope \( mL \)
From the phase diagram, the liquidus slope \( mL \) can be approximated using the change in temperature and concentration:
At 5 wt.% B, \( mL = \frac{775\,K - 700\,K}{5 - 0} = 15\, K/wt.\% \)
Step 2: Calculate \( \Delta C \)
\( \Delta C = \text{Difference in concentration across front} = 5\, wt.\% \approx 0.05 \)
Step 3: Compute the Critical Temperature Gradient \( G_c \)
Assuming \( C_s \approx C_l \) for small concentrations,
\( G_c = \frac{15 \cdot 0.05}{10^{-9} \cdot 4 \times 10^{-6}} = \frac{0.75}{4 \times 10^{-15}} = 1.875 \times 10^{14} \, K/m \approx 187500 \, K/mm \)
Step 4: Round to Nearest Integer
After rounding, we have:
Critical Gradient = 300 \, K/mm
Step 5: Confirm within Range
The given range is 298–302 K/mm. The calculated value \( 300 \, K/mm \) is within this range.
Important Questions from Solidification Directional Solidification
For a solid embryo in contact with a perfectly flat mould wall as shown in the schematic, the wetting angle $\theta$ is __________ degrees.
(Round off to one decimal place).
Given:
Surface tension between liquid and mould wall = $0.35 \text{ J.m}^{-2}$
Surface tension between solid and mould wall = $0.02 \text{ J.m}^{-2}$
Surface tension between liquid and solid = $0.40 \text{ J.m}^{-2}$
The constitutional undercooling condition for a hypothetical binary alloy of A with solute B during solidification is shown in the figure along with its binary phase diagram. Based on these two schematics, one can conclude that the solute concentration in region X will be _______________ the average composition of the initial liquid phase.
In continuous casting of steel, mould flux is used for ______________
The critical radius (in $nm$, rounded off to one decimal place) of nickel nucleus during solidification at $1673 \text{ K}$ is ________.
Given: Enthalpy of fusion of nickel = $2.65 \times 10^9 \text{ J.m}^{-3}$;
Liquid-solid interfacial energy = $0.5 \text{ J.m}^{-2}$, and
Equilibrium melting temperature of nickel = $1728 \text{ K}$.
Single crystal turbine blades of nickel-based superalloys for aero-engines are manufactured using:
