In this work, we adopt the so called "dust mantle model", in which a dry dust layer is present at the surface of the nucleus over the dust-ice mixture and the water ice sublimation occurs at the boundary between the mantle and the ice front (Hu et al. 2017) (and references therein).
2.1.

Dust mantle model The nucleus is regarded as a polyhedron consisting of triangular facets.

For each facet, neglecting the changes in thermal properties due to the gas from ice sublimation filling the porosity of the nucleus, the 1D heat conduction equation can be expressed as follows:
∂tT = κ ∂2
ρC ∂
∂x2 T, (1) where ρ, C, κ respectively correspond to the density, specific heat capacity and thermal conductivity.

They are assumed to be different constants in dry dust mantle and dust-ice mixture, denoted as ρd, Cd, κd and ρm, Cm, κm.

In fact, ρmCm, κm can be expressed in terms of ρd, Cd, κd and ρi, Ci, κi of the pure water ice (Davidsson & Skorov 2002; Davidsson & Gutiérrez 2005):
ρmCm = (1 −f)ρdCd + fρiCi, κm = h((1 −f)κd + fκi) (2) where f is the icy area fraction, h is the Hertz factor dependent on the morphological structure of the medium and is generally regarded as a free parameter (Davidsson & Skorov 2002).

The surface boundary condition of the dust mantle derived from conservation of energy, reads (Kührt & Keller 1994; Lagerros 1996) ∂ ∂xTx=0 = (1 −AB)sψF⊙, (3)
ϵσT 4 x=0 −κd where ϵ is the emissivity, σ is the Stefan–Boltzmann constant. (1 −AB)sψF⊙describes the radiation flux received by the facet, AB is the Bond albedo, s indicates whether the facet is illuminated by the Sun, considering the horizon shadows and projected shadows, ψ represents the cosine function of the solar elevation
angle and F⊙is the incident solar radiation flux determined by F⊙= S ⊙/r2, where r is the heliocentric distance of the asteroid and S ⊙is the solar constant.

It is necessary to introduce an additional boundary condition at the interface between the dust mantle and dust-ice mixture, due to the energy consumption caused by water ice sublimation.

Assuming the thickness of the dust mantle is X, the condition can be expressed as (Kührt & Keller 1994; Hu et al. 2017):
∂
∂xTx=X+ + lZ(X),
−κd ∂
∂xTx=X−= −κm
(4) l is the latent heat of water ice, Z(X) is the mass flux of sublimation determined by (Hu et al. 2017)
Z(X) = ΨfZH−K(T(X)),
(5) where � ˆm 2πkBT , (6)
ZH−K(T) = α(T)PV(T)
is the Hertz-Knudsen formula, describing the sublimation flux from the surface of pure solid ice into vacuum (Kossacki et al. 1999; Gundlach et al. 2011). ˆm is the mass of a water molecules and kB is the Boltzmann constant.

PV(T) is the saturation vapor pressure. α(T) is the sublimation coefficient, representing the reduction in sublimation flux due to the reattachment of gas molecules to the ice surface after impacting with each other.

They are respectively evaluated as (Panale & Salvail 1984; Mauersberger & Krankowsky 2003; Gundlach et al. 2011) PV(T) = a exp (−b
T ), α(T) = c0 + c1 1 + exp (c2 −c3/T), (7)
with constants a, b, c0, c1, c2, c3 taking the values 3.23×1012 Pa, 6134.6 K, 0.146, 0.854, 57.78, 11580 K.

The parameter Ψ in Eq. (5) represents the reduction factor in the sublimation flux from the bare icy surface, accounting for the permeability of the mantle to gas flow.