3D image-based numerical simulations and experimental measurements of the thermal conductivity of snow

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

²

and 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

^-1

K

^-1

Thermal conductivity k

F _z

Atmosphere, T

₁



^z

T

Ground, T

₂

Thermal 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

^{*  }

^z

Sturm 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







_α

α

k

F

 T



 k

_eff

F

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

_eff

Solving 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







_α

α

k

F

 T



 k

_eff

F

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

_eff

Precipitation Particles Depth Hoar

REV

• Smallest volume from which a variable representative of the whole can be estimated

• Computation of k

_eff

on volume at least equal to the REV

• REV estimated by calculating k

_eff

on increasing volumes

Results: k

_eff

versus density

• Good correlation between k

_eff

and

density.

• Anisotropy of k

_eff

= vertical component horizontal component

 Anisotropy range:

0.7 – 1.5

Results: k

_eff

versus 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

_eff

versus 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

^-1

K

^-1

.

• Even if k

_air

<< k

_ice

 air plays a vital role in heat conduction

• Kaempfer et al., 2005: k

_air

neglected  underestimation?

Influence of conduction in air

- 90%

- 40%

- 30%

Comparison of k estimations

Method Average value of k (W m

^-1

K

^-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

^*_needle

values are significantly lower than computed values.

Comparison of k estimations

• k

^*_needle

values are significantly lower than computed values.

1) Influence of the temperature?

Sturm et al. 1997: measurements at low temperatures.

 k

_eff

increases 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

3 different estimations:

 Needle probe - assumes a homogeneous and isotropic medium - transient mode (strong local temperature gradients)

- all thermal processes

 Flux-gradient - vertical component of k*

- steady state condition - all thermal processes

 Numerical computation - full tensor of k

_eff

- pure conduction

Differences between these 3 variables may depend on the snow type and on temperature and temperature gradient because of the non- conductive processes.

Why needle-probe underestimates the thermal

conductivity compared to other methods?

Conclusions

• k

_eff

strongly correlated with snow density (Yen,1981)

support the use of the k-density relationship in snowpack models

• Anisotropy of k

_eff

linked to snow microstructure (0.7-1.5)

• Air plays an important role in heat conduction

• Underestimation of k

^*

with the needle probe

Perspectives

Better understanding of non-conductive processes

• influence of the snow type, temperature and T. gradient

• contribution to total heat transfer

• impact on the experimental methods

your attention

Acknowledgements:

Meteo-France and INSU-LEFE. J.

Baruchel, E. Boller, W. Ludwig, X.

Thibault (ESRF), P. Charrier, J.

Desrues (3S-R), CEN staff and F.

Dominé (LGGE).

Paper in press: Calonne, N., F. Flin, S. Morin, B. Lesaffre, S. Rolland

du Roscoat, and C. Geindreau (2011), Numerical and experimental

investigations of the effective thermal conductivity of snow,

Geophys. Res. Lett., doi:10.1029/2011GL049234.