This presentation illustrates the principles of thermodynamics in the freezing soil according to the capillary schematization and the freezing=drying assumption

1. The thermodynamics of
freezing soils
Matteo Dall’Amico(1), Riccardo Rigon(1), Stephan Gruber(2)
and Stefano Endrizzi(3)
Vienna, 5 may 2010
(1) Department of Environmental engineering, University of Trento, Trento, Italy (matteo.dallamico@ing.unitn.it)
(2) Department of Geography, University of Zurich, Switzerland
(3) National Hydrology Research Centre, Environment Canada, Saskatoon, Canada,
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2. Phase transition in soil
How do we model the liquid-
solid phase transition in a soil?
What are the assumptions behind the heat equation?
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3. Back to fundamentals...
Uc ( ) := Uc (S, V, A, M ) Internal Energy
entropy interfacial area Independent extensive
volume mass variables
dUc (S, V, A, M ) ∂Uc ( ) ∂S ∂Uc ( ) ∂V ∂Uc ( ) ∂A ∂Uc ( ) ∂M
= + + +
dt ∂S ∂t ∂V ∂t ∂A ∂t ∂M ∂t
temperature pressure surface chemical Independent intensive
energy potential variables
dUc (S, V, A, M ) = T ( )dS − p( )dV + γ( ) dA + µ( ) dM
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4. Clausius-Clapeyron relation
Gibbs-Duhem identity:
SdT ( ) − V dp( ) + M dµ( ) ≡ 0
hw ( ) hi ( )
− dT + vw ( )dp = − dT + vi ( )dp
T T
Equilibrium condition:
dµw (T, p) = dµi (T, p)
water ice
dp hw ( ) − hi ( ) Lf ( )
= ≡
dT T [vw ( ) − vi ( )] T [vw ( ) − vi ( )]
p: pressure [Pa]
T: temperature [˚C]
s: entropy [J kg-1 K-1]
h: enthalpy [J kg-1]
v: specific volume [m3 kg-1]
Lf = 333000 [J kg-1] latent
heat of fusion
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5. References on thermodynamic equilibrium
look similar but are
actually different...
they claim to use
the Clausius-
Clapeyron
relation but...
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6. Clausius-Clapeyron relation
Gibbs-Duhem identity:
SdT ( ) − V dp( ) + M dµ( ) ≡ 0
hw ( ) hi ( )
− dT + vw ( )dp = − dT + vi ( )dp
T T
Equilibrium condition:
dµw (T, p) = dµi (T, p)
water ice
dp hw ( ) − hi ( ) Lf ( )
= ≡
dT T [vw ( ) − vi ( )] T [vw ( ) − vi ( )]
p: pressure [Pa]
T: temperature [˚C]
s: entropy [J kg-1 K-1]
h: enthalpy [J kg-1] ????
v: specific volume [m3 kg-1]
Lf = 333000 [J kg-1] latent
heat of fusion
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7. Capillary schematization
Suppose an air-water interface. The Young-Laplace
equation states the pressure relationship:
∂Awa (r) ∂Awa /∂r 2
pw = pa − γwa = pa − γwa = pa − γwa := pa − pwa (r)
∂Vw (r) ∂Vw /∂r r
pa
pw
pi Suppose an ice-water interface. The 2nd principle of
thermodynamics sets the equilibrium condition:
1 1 pw + γiw ∂Aiw
∂Vw pi µw µi
dS = − dUw + − dVw − − dMw = 0
Tw Ti Tw Ti Tw Ti
therefore:

 Ti = Tw p: pressure [Pa]
pi = pw + γiw ∂Aiw A: surface area [m2]
 ∂Vw γ: surface tension [N m-1]
µi = µw r : capillary radius [m] 7
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8. Two phases interfaces
Suppose an air-ice and a ice-water interface:
∂Aia r(0) ∂Aiw (r1 )
pw1 = pa − γia − γiw
∂Vw ∂Vw
Two interfaces should be considered!!!
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9. “Freezing=drying” assumption
Considering the assumption
“freezing=drying” (Miller, 1963, pi=pa
Spaans and Baker, 1996) the ice γia = γwa = γiw
“behaves like air”:
saturation
degree ∂Awa r(0) ∂Awa (r1 )
pw1 = pa − γwa − γwa
∂Vw ∂Vw
{
{
pw0 ∆pfreez
air-water interface water-ice interface
saturation degree freezing degree
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10. The freezing process
From the Gibbs-Duhem equation on obtains the
Generalized Clapeyron equation:
Lf
hw ( ) hi ( ) ∆pf reez ≈ ρw (T − T0 )
− dT + vw ( )dpw = − dT + vi ( )dpi T0
T T
Freezing pressure:
Lf
big pores pw1 ≈ pw0 + ρw (T − T0 )
T0
saturation medium pores
degree
Depressed freezing point:
g T0
T := T0 +
∗
ψ w0
Lf
small pores
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11. Freezing schematization
Unsaturated Freezing
unfrozen starts
Unsaturated Freezing
Frozen procedes
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12. Freezing schematization
Unsaturated Freezing
unfrozen starts
Unsaturated Freezing
Frozen procedes
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13. Soil Freezing Characteristic curve (SFC)
“freezing=drying” assumption allows to “exploit” the theory of unsaturated soils:
Unfrozen water content: pressure head:
pw
θw (T ) = θw [ψw (T )] ψw =
ρw g
soil water retention curve + Clausius Clapeyron
e.g. Van Genuchten (1980)
soil suction psi
0.4
0
ψw0
0.3
ψfreez
soil suction psi [m]
theta_w [-]
-5
0.2
-10
0.1
T*
0.0
-15
-0.10 -0.05 0.00 0.05 0.10
-10000 -8000 -6000 -4000 -2000 0
Temperature [ C]
Psi [mm] psi_m=-1m - Tstar= -0.008
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14. ψw0
Soil Freezing...Characteristic curve (SFC)
Unfrozen water content
0.4
ψw0
psi_m −5000
ψw0
psi_m −1000
0.3 ψw0
psi_m −100
ψw0
psi_m 0 air
Theta_u [−]
0.2
ice
0.1
depressed
water
melting point
−0.05 −0.04 −0.03 −0.02 −0.01 0.00
temperature [C]
n −m
Lf
θw = θr + (θs − θr ) · 1 + −αψw0 − α (T − T ∗ ) · H(T − T ∗ )
g T0
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15. Freezing schematization with SFC
Unsaturated θw θw Freezing
unfrozen starts
Unsaturated θw θw Freezing
Frozen procedes
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16. Energy conservation
The heat equation below written
hides important hypothesis, often
tacitly assumed:
apparent
temperature water flux
heat capacity
[˚C] [m s-1]
[J m-3 K -1]
Harlan (1973)
∂T Jw T ∂ ∂T Guymon and Luthin (1974)
Ca + ρw cw = λ Fuchs et al. (1978)
Zhao et al. (1997)
∂t ∂z ∂z ∂z Hansson et al. (2004)
Daanen et al. (2007)
Watanabe (2008)
mass heat
water
capacity thermal
density
[J kg-1 K -1] conductivity
[kg m-3]
[W m-1 K -1]
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17. Energy conservation
ph
U = hg Mg + hw Mw + hi Mi − (pw Vw + pi Vi ) + ph
µw Mw + µi Mi
0 assuming equilibrium thermodynamics: µw=µi
and Mwph = -Miph
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18. Energy conservation
ph
U = hg Mg + hw Mw + hi Mi − (pw Vw + pi Vi ) + ph
µw Mw + µi Mi
0 assuming equilibrium thermodynamics: µw=µi
and Mwph = -Miph
0 assuming freezing=drying
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19. Energy conservation
ph
U = hg Mg + hw Mw + hi Mi − (pw Vw + pi Vi ) + ph
µw Mw + µi Mi
0 assuming equilibrium thermodynamics: µw=µi
and Mwph = -Miph
no water flux during phase
0 assuming: change (closed system) 0 assuming freezing=drying
no volume expansion: ρw=ρi
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20. Energy conservation
ph
U = hg Mg + hw Mw + hi Mi − (pw Vw + pi Vi ) + ph
µw Mw + µi Mi
0 assuming equilibrium thermodynamics: µw=µi
and Mwph = -Miph
no water flux during phase
0 assuming: change (closed system) 0 assuming freezing=drying
no volume expansion: ρw=ρi
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21. Energy conservation
G = −λT (ψw0 , T ) · T conduction
∂U
+ • (G + J) + Sen = 0
∂t J = ρw · Jw (ψw0 , T ) · [Lf + cw T ] advection
• no water flux during phase change (closed system)
• freezing=drying
• no volume expansion (ρw=ρi)
U = CT · T + ρw Lf θw
Lf closure relation
∆pf reez ≈ ρw (T − T0 )
T0
CT := Cg (1 − θs ) + ρw cw θw T + ρi ci θi
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22. Energy conservation
G = −λT (ψw0 , T ) · T conduction
∂U
+ • (G + J) + Sen = 0
∂t J = ρw · Jw (ψw0 , T ) · [Lf + cw T ] advection
• no water flux during phase change (closed system)
• freezing=drying
• no volume expansion (ρw≠ρi)
U = CT · T + ρw [Lf − ψw g] θw
Lf closure relation
∆pf reez ≈ ρw (T − T0 )
T0
CT := Cg (1 − θs ) + ρw cw θw T + ρi ci θi
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23. Energy conservation
G = −λT (ψw0 , T ) · T conduction
∂U
+ • (G + J) + Sen = 0
∂t J = ρw · Jw (ψw0 , T ) · [Lf + cw T ] advection
• no water flux during phase change (closed system)
• freezing=drying
• no volume expansion (ρw≠ρi)
U = CT · T + ρw [Lf − (ψw − ψi ) g] θw
Lf pw pi
(T − T0 ) = − closure relation
T0 ρw ρi Christoffersen and Tulaczyk (2003)
CT := Cg (1 − θs ) + ρw cw θw T + ρi ci θi
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24. Conclusions
1. The assumption “freezing=drying” (Miller, 1963) is a convenient hypothesis
that allows to get rid of pi and find a closure relation.
2.The common heat equation with phase change used in literature implies
that there is no work of expansion from water to ice and that water density
is equal to ice density.
3. The “freezing=drying” assumption is limitating to model phenomena like
frost heave. In this case, a more complete approach should be used where
also the ice pressure is fully accounted (Rempel et al. 2004, Rempel, 2007, Christoffersen
and Tulaczyk, 2003).
4. The thermodynamic approach of the freezing soil allows to write the set of
equations according to the particular problem under analysis.
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25. Thank you!
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