and experimental measurements of the thermal conductivity of snow
N. Calonne
1, F. Flin
1, S. Morin
1, B. Lesaffre
1, S. Rolland du Roscoat
2and C. Geindreau
2.
1
Meteo-France - CNRS, CNRM-GAME URA 1357, CEN, Grenoble, France
2
3S-R CNRS UMR 5521, Universite Joseph Fourier - Grenoble INP, Grenoble, France
1 mm
T
k F
Heat transfer and thermal conductivity
Fourier’s law:
Heat transfer:
- Conduction in ice - Conduction in air
- Phase change effects
- Air convection in the pores - Water vapor diffusion
Effective thermal conductivity: k
eff- from computations - only conduction
Apparent thermal conductivity: k
*- from measurements - all processes
- scalar (isotropic medium or 1D) where k is a tensor, in W m
-1K
-1Thermal conductivity k
F z
Atmosphere, T
1
zT
Ground, T
2Thermal conductivity in snowpack models
Yen, 1981
Sturm et al., 1997
Experimental fits Experimental data
k
*= f ( density )
Accepted idea:
large scatter
Issues and contributions
Need
- Re-assessment of the variables related to conductivity
- Partitioning between physical phenomena involved in heat transfer
- Assessment of the experimental methods
Our study
Numerical simulations of conductivity from 3D snow microstructure provided by tomography (e.g. Kaempfer et al. 2005).
Innovations:
- conduction in ice and air was simulated - access to the full tensor of k
eff- on a wide range of seasonal snow types (30 samples)
5
Examples of snow samples (out of 30)
Precipitation Particles
Decomposing and
Fragmented precipitation particles
Rounded
Grains Depth
Hoar
1 mm 1 mm
1 mm
Melt Forms
1 mm 1 mm
Cold-room experiment
Estimation of conductivity by 3 different methods:
- Needle probe - Flux-gradient
- Computation from 3D images
Insulating polystyrene plate Copper plate
Snow slab Thermal regulation of the copper plates by a fluid circulation
T = - 1 °C T = - 7 °C
|| T|| = 43 K m-1 1 m
14 cm
Needle probe method
Sturm et al. 1997, Morin et al.
2010, Riche and Schneebeli 2010.
- Monitors the temperature rise during the heating time.
- In transient mode
- Includes all thermal processes - Assumes a homogeneous and isotropic medium
k
*needle(scalar)
Flux-gradient method
Heat-flux sensor Temperature probe
- In steady state condition
- Includes all thermal processes - Provides vertical component of k
* k
*z(scalar)
ΔT
F Δz
k
z*
zSturm et al. 1997, Morin et al.
2010, Riche and Schneebeli 2010.
Schneebeli and Sokratov 2004, Satyawali et al. 2008.
Needle probe method
- Monitors the temperature rise during the heating time.
- In transient mode
- Includes all thermal processes - Assumes a homogeneous and isotropic medium
k
*needle(scalar)
1.6 cm 5.35 cm
1. Sampling 4. Machining
3. Freezing 2. Impregnation
Numerical computation - Acquisition of 3D images
5. X-ray tomography 6. Image processing
1 mm
- Image size: 2.5 - 9.5 mm
- Voxel size: 5 - 10 µm
L
Snow layer
Macroscopic scale Microscopic scale
T
α
αα
k
F
T
k
effF
ice) air, ( α
Condition
Separation of scale L >> l
l
Numerical computation - Homogenisation
Auriault et al. 2009
Periodic homogenisation
k
eff(micros., k
air, k
ice)
zz zy
zx
yz yy
yx
xz xy
xx
k k
k
k k
k
k k
k
k
effSolving a specific boundary problem arising from the homogenisation process, on a 3D image, by Geodict software.
Numerical computation - Homogenisation
L
Snow layer
Macroscopic scale Microscopic scale
T
α
αα
k
F
T
k
effF
ice) air, ( α
l
Auriault et al. 2009
Periodic homogenisation
k
eff(micros., k
air, k
ice)
0,1 0,2 0,3 0,4 0,5 0,6 0,7
0 1 2 3 4 5 6
Size (mm) keff (W m-1 K-1 )
kxky kz
0 0,05 0,1 0,15 0,2
0 1 2 3 4
Size (mm) keff (W m-1 K-1 )
kx ky kz
REV
Representative Elementary Volume (REV) for k
effPrecipitation Particles Depth Hoar
REV
• Smallest volume from which a variable representative of the whole can be estimated
• Computation of k
effon volume at least equal to the REV
• REV estimated by calculating k
effon increasing volumes
Results: k
effversus density
• Good correlation between k
effand
density.
• Anisotropy of k
eff= vertical component horizontal component
Anisotropy range:
0.7 – 1.5
Results: k
effversus density
• Anisotropy is linked to snow microstructure:
- horizontal: some natural snow (depending on the type of snowfall, settling…) .
- vertical: snow exposed to a vertical temperature gradient.
Results: k
effversus density
Snow
type Density
(kg m-3)
Vertical component of k
eff (W m-1K-1)with air without air
+ PP 103 0.06 0.006
RG 256 0.16 0.10
^ DH 315 0.25 0.18
• At –2°C, k
air= 0.024 and k
ice= 2.107 W m
-1K
-1.
• Even if k
air<< k
ice air plays a vital role in heat conduction
• Kaempfer et al., 2005: k
airneglected underestimation?
Influence of conduction in air
- 90%
- 40%
- 30%
Comparison of k estimations
Method Average value of k (W m
-1K
-1)
Vertical component Isotropic assumption
Flux-gradient 0.240 -
Computation normalized 0.238 0.203
Needle probe - 0.158
6 estimations of k from our TG experiment
by 3 differents methods
on same snow samples
• k
*needlevalues are significantly lower than computed values.
Comparison of k estimations
• k
*needlevalues are significantly lower than computed values.
1) Influence of the temperature?
Sturm et al. 1997: measurements at low temperatures.
k
effincreases with decreasing temperature.
2) Problem with needle probe parameters?
Sturm et al.1997: Δt
heating=120s et ΔQ such as ΔT = +2-4K.
OK according to Morin et al. 2010.
3) Bias linked to microstructural disturbance?
Riche and Schneebeli 2010
Morin et al. 2010: no signifiant effects on k
*provided that the first 30s of heating are discarded.
Need to a specific study on this subject.
Why does needle-probe underestimate the thermal conductivity compared to other methods?
Riche and Schneebeli 2010