Phase Change Problems With Temperature-Dependent Thermal Conductivity

1974 ◽  
Vol 96 (2) ◽  
pp. 214-217 ◽  
Author(s):  
S. H. Cho ◽  
J. E. Sunderland
Keyword(s):  
Thermal Conductivity ◽  
Exact Solution ◽  
Phase Change ◽  
The Body ◽  
Temperature Dependent ◽  
Fusion Temperature ◽  
Infinite Body ◽  
A Value ◽  
Change Position ◽  
Temperature Dependent Thermal Conductivity

An exact solution is presented for the temperature distribution and phase change position of a semi-infinite body which is initially at a constant temperature different from the fusion temperature. The temperature of the free surface is instantaneously changed and held constant at a value that will cause either melting or solidification of the body. The thermal conductivity is assumed to vary linearly with temperature. The convection in the liquid phase due to density change is also considered.

Stagnant lid convection with temperature-dependent thermal conductivity and the thermal evolution of icy worlds

2020 ◽  
Vol 224 (3) ◽  
pp. 1870-1889
Author(s):  
Frédéric Deschamps
Keyword(s):  
Thermal Conductivity ◽  
Thermal Evolution ◽  
Heat Fluxes ◽  
The Body ◽  
Temperature Dependent ◽  
Temperature Dependent Conductivity ◽  
Icy Bodies ◽  
Main Component ◽  
The Impact ◽  
Temperature Dependent Thermal Conductivity

SUMMARY Convection is an efficient process to release heat from planetary interiors, but its efficiency depends on the detailed properties of planetary mantles and materials. A property whose impact has not yet been studied extensively is the temperature dependence of thermal conductivity. Because thermal conductivity controls heat fluxes, its variations with temperature may alter heat transfer. Here, I assess qualitatively and quantitatively the influence of temperature-dependent thermal conductivity on stagnant lid convection. Assuming that thermal conductivity varies as the inverse of temperature $(k \propto 1/T)$, which is the case for ice Ih, the main component of outer shells of solar System large icy bodies, I performed numerical simulations of convection in 3-D-Cartesian geometry with top-to-bottom viscosity and conductivity ratios in the ranges 105 ≤ Δη ≤ 108 and 1 ≤ Rk ≤ 10, respectively. These simulations indicate that with increasing Rk, and for given values of the Rayleigh number and Δη, heat flux is reduced by a factor Rk0.82, while the stagnant lid is thickening. These results have implications for the structures and thermal evolutions of large icy bodies, the impact of temperature-dependent conductivity being more important with decreasing surface temperature, Tsurf. The heat fluxes and thermal evolutions obtained with temperature-dependent conductivity are comparable to those obtained with constant conductivity, provided that the conductivity is fixed to its value at the bottom or in the interior of the ice shell, that is, around 2.0–3.0 W m−1 K−1, depending on the body. By contrast, temperature-dependent conductivity leads to thicker stagnant lids, by about a factor 1.6–1.8 at Pluto (Tsurf = 40 K) and a factor 1.2–1.4 at Europa (Tsurf = 100 K), and smaller interior temperatures. Overall, temperature-dependent thermal conductivity therefore provides more accurate descriptions of the thermal evolutions of icy bodies.

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