Solidification Morphology and Bifurcation Predictions with the Maximum Entropy Production Rate Model

. 2019 Dec 26;22(1):40. doi: 10.3390/e22010040

Letter symbols
Af	area of a solute flux in a liquid (m²)
A_SLI	area of an interface in a solid–liquid region (m²)
B	Entropic contribution from the solute boundary layer
Cp	average heat capacity across a solid–liquid interface (Jm⁻³K⁻¹)
d	interplanar lattice spacing (m)
$Δ C_{O}$	change in concentration at a solute distance z (mole m⁻³) $Δ C_{O} = \frac{Δ T_{O}}{m_{L}}$
D	Diffusion Coefficient (m²s⁻¹)
fs	fraction of liquid solidified at the solid–liquid interface (dimensionless)
G_S	temperature gradient in a solid (Km⁻¹)
G_L	temperature gradient in a liquid (Km⁻¹)
G_SLI	linear temperature gradient across a diffuse interface (Km⁻¹)
Δh_m	heat of fusion of a solid with defects (Jm⁻³)
Δh_sl	equilibrium heat of fusion (Jm⁻³) or (Jmol⁻¹)
Js	solute flux in a liquid entering a solid–liquid interface (mole s⁻¹)
k	equilibrium partition coefficient obtained from the phase diagram (dimensionless)
k_eff	effective partition coefficient at a solid–liquid interface (dimensionless)
ΔKE	gain or loss in kinetic energy (J)
KL	thermal conductivity for a rigorous liquid (Jm⁻¹K⁻¹s⁻¹)
KS	thermal conductivity for a rigorous solid (Jm⁻¹K⁻¹s⁻¹)
m_L	slope of the equilibrium liquidus line at the SLI for a binary material (Km³mole⁻¹)
M	Entropic contribution from the thermal flow
N	Diffuseness contribution to the entropy generation
Q	lost work potential from the heat generation from a solid–liquid interface (J)
R_g	molar gas constant (Jmol⁻¹ K⁻¹)
S	Mullins and Sekerka stability constant (dimensionless)
S_f	flux entropy rate (JK⁻¹s⁻¹)
${\dot{s}}_{LG}$	entropy generation density due to solute gradient in a liquid (Jm⁻³K⁻¹)
${\dot{s}}_{SG}$	entropy generation density due to solute gradient in a solid (Jm⁻³K⁻¹)
${\dot{s}}_{E}$	change in entropy generation rate density due to exchange of matter and energy to and from a solid–liquid interface with its surrounding (Jm⁻³K⁻¹s⁻¹)
${\dot{s}}_{gen}$	irreversible entropy generation rate in a diffuse region (JK⁻¹s⁻¹)
${\dot{s}}_{in}$	rate of entropy entering a control volume (JK⁻¹s⁻¹)
${\dot{s}}_{out}$	rate of entropy leaving a control volume (JK⁻¹s⁻¹)
${\dot{s}}_{gen}$	total irreversible entropy generated rate density at an interface (Jm⁻³K⁻¹)
${\dot{s}}_{LG}$	entropy generation rate density by the solute gradient in a liquid (Jm⁻³K⁻¹)
(𝑆𝑔𝑒𝑛)𝑚𝑎𝑥	maximum entropy generation due to lost work (JK⁻¹)
𝑑𝑆_𝑐𝑣⁄𝑑𝑡	total steady state entropy rate in a control volume (JK⁻¹s⁻¹)
𝑑𝑠_𝑐𝑣⁄𝑑𝑡	total steady state entropy rate density in a control volume (Jm⁻¹K⁻¹s⁻¹)
t	time (s)
Tli	liquidus temperature at a solid–liquid interface boundary (K)
Tsi	solidus temperature at a solid–liquid interface boundary (K)
ΔT_SLI	temperature difference across a solid–liquid interface (K)
(𝑑𝐶𝐿𝐺⁄𝑑𝑧) or (𝛥𝐶𝑂⁄𝛿𝑐)	change in solute gradient in a liquid (mole m⁻⁴)
Tm	melting temperature (K)
Tav	average temperature between Tli and Tsi across a diffuse interface (K)
ΔT_O	solidification temperature range (K)
V	solidification interface velocity (ms⁻¹)
W_L	lost work (J)
dz or δc	change in the position length of the solute (m)
Z_CUT	deviation parameter of CUT from experiment at breakdown (dimensionless)
Z_LST	deviation parameter of LST from experiment at breakdown (dimensionless)
Greek symbols
Ωf	flux volume (m³)
ΔΩS	volume shrinkage (m³)
$δ_{C}$	Solute boundary layer at the critical point $δ_{C} = \frac{2 D_{L}}{V}$ (m)
\|Δρk\|	density shrinkage (kgm⁻³) (ρs- ρl)
ρl	density of rigorous liquid (kgm⁻³)
ρs	density of rigorous solid (kgm⁻³)
Δμ_c	Chemical potential difference (Jmole⁻¹)
ζ	solid–liquid interface thickness (m)
ω_D	energy of defects (Jm⁻³)
Ω_SLI	volume of a solid–liquid interface (m³)
$\dot{φ}$	maximum entropy generation rate density for a moving interface (Jm⁻³K⁻¹s⁻¹)
η_G	driving force diffuseness (dimensionless)
η_T	total diffuseness (dimensionless)
𝜂_𝛼	thermal diffuseness (dimensionless)
Subscripts and acronyms
CUT	constitutional undercooling theory circa 1953
LST	linear stability theory circa 1964
MEPR	maximum entropy production rate circa 2011
L	liquid
S	solid
LG	solute gradients in the liquid
SG	solute gradients in the solid
SLI	solid–liquid interface
HD	mean heat dissipation at the solid–liquid interface
f	facet
nf	non-facet
Expt	experiment