Traditionally two types of equations for representing frost-pillars have been used: One type is based on considering heat conduction through the pillars, while the other is considering only the effect of long wave radiation at the surface of the frost-pillars. The former is inappropriate for describing short pillars of frost, while the shortcoming of latter is inadequate description of the long frost-pillars.
Unifying above two equations, we intended to derive one that could apply to both long and short frost-pillars. In this study, we have taken into consideration the heat balances that occur at both the pillars' surface and base, and from the viewpoint of unifying both, have derived the following equation:
dl/dt=A
3l+A
4/A
2l+k
iwhere
A
1=-θ
a(h+4ε
lσT
a3+L
ih
va′)+ε
lR
n0+L
ih
vζρ
asA
2=h+4ε
lR
n0/T
0+4ε
1σT
a3+L
ih
va′
A
3=A
1h′
va′+A
0A
2A
4=k
i(A
1/Lρ
iα+A
0)
A
0=h′
v(a′θ
a-ζρ
as)-q
s0/Lρ
iα
k
i=αk
i+(1-α)k
awhere d
l/d
t; growing velocity of frost columns, θ
a,
Ta; air temperature (°C), (°K),
h; heat-transfer coefficient, ε
l; emissivity of long wave radiation from surface of frost columns, σ; Stefan-Bolzmann constant,
Li; latent heat of ice sublimation,
hv; transfer coefficient of water vapor,
a′; dρ/d
T,
Rn0; net radiation of black body at 0°C, ζ; (1-ζ)=relative humidity of air, ρ
as; density of water vapor saturation at a given temperature,
T0; 0°C (°K),
h′
v;
hv/ρ
i⋅ρ
i; density of ice,
qs0; heat flux from ground,
L; latent heat of water freezing,
ki; heat conductivity of ice, and
ka; heat conductivity of air between columns of ice.
The existing two equations can be obtained as approximations by simplifying given conditions of our equation.
The calculated values obtained by our equation correspond values of both short and long frost-pillars.
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