Thermodynamics of freezing soil

1

Matteo Dall’Amico(1), Riccardo Rigon(1), Stephan Gruber(2) and Stefano Endrizzi(3)

(1) Department of Environmental engineering, University of Trento, Trento, Italy ([email protected])(2) Department of Geography, University of Zurich, Switzerland(3) National Hydrology Research Centre, Environment Canada, Saskatoon, Canada,

The thermodynamics offreezing soils

Vienna, 5 may 2010

Tuesday, May 11, 2010

mailto:[email protected]

mailto:[email protected]

Matteo Dall’Amico et al, EGU 2010

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?


Uc( ) := Uc(S, V, A, M)

dUc(S, V, A, M)dt

=!Uc( )

!S

!S

!t+

!Uc( )!V

!V

!t+

!Uc( )!A

!A

!t+

!Uc( )!M

!M

!t


Back to fundamentals...

3

Internal Energy

entropy interfacial areavolume mass

Independent extensive variables

dUc(S, V, A, M) = T ( )dS ! p( )dV + !( ) dA + µ( ) dM

temperature pressure surface energy

chemical potential

Independent intensive variables


SdT ( )! V dp( ) + Mdµ( ) " 0

dµw(T, p) = dµi(T, p)

!hw( )T

dT + vw( )dp = !hi( )T

dT + vi( )dp

dp

dT=

hw( )! hi( )T [vw( )! vi( )]

" Lf ( )T [vw( )! vi( )]


Clausius-Clapeyron relation

4

Gibbs-Duhem identity:

Equilibrium condition:

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

water ice



5

References on thermodynamic equilibrium

they claim to use the Clausius-Clapeyronrelation but...

look similar but are actually different...


SdT ( )! V dp( ) + Mdµ( ) " 0

dµw(T, p) = dµi(T, p)

!hw( )T

dT + vw( )dp = !hi( )T

dT + vi( )dp

dp

dT=

hw( )! hi( )T [vw( )! vi( )]

" Lf ( )T [vw( )! vi( )]


Clausius-Clapeyron relation

6

Gibbs-Duhem identity:

Equilibrium condition:

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

water ice

????


pw = pa ! !wa"Awa(r)"Vw(r)

= pa ! !wa"Awa/"r

"Vw/"r= pa ! !wa

2r

:= pa ! pwa(r)

dS =!

1Tw

! 1Ti

"dUw +

#pw + !iw

!Aiw!Vw

Tw! pi

Ti

$dVw !

!µw

Tw! µi

Ti

"dMw = 0

!"

#

Ti = Tw

pi = pw + !iw!Aiw!Vw

µi = µw


7

pa

pw

Suppose an ice-water interface. The 2nd principle of thermodynamics sets the equilibrium condition:

therefore:

Capillary schematization

p: pressure [Pa]A: surface area [m2]γ: surface tension [N m-1]r : capillary radius [m]

pi

Suppose an air-water interface. The Young-Laplace equation states the pressure relationship:



Two phases interfaces

8

pw1 = pa ! !ia"Aiar(0)

"Vw! !iw

"Aiw(r1)"Vw

Two interfaces should be considered!!!

Suppose an air-ice and a ice-water interface:


pw1 = pa ! !wa"Awar(0)

"Vw! !wa

"Awa(r1)"Vw


“Freezing=drying” assumption

9

Considering the assumption “freezing=drying” (Miller, 1963, Spaans and Baker, 1996) the ice “behaves like air”:

pi=paγia = γwa = γiw

saturationdegree {

pw0{∆pfreez

air-water interfacesaturation degree

water-ice interfacefreezing degree


!pfreez ! !wLf

T0(T " T0)

T ! := T0 +g T0

Lf!w0


The freezing process

10

saturationdegree

!hw( )T

dT + vw( )dpw = !hi( )T

dT + vi( )dpi

pw1 ! pw0 + !wLf

T0(T " T0)

From the Gibbs-Duhem equation on obtains the Generalized Clapeyron equation:

big pores

medium pores

small pores

Freezing pressure:

Depressed freezing point:



11

Unsaturatedunfrozen

UnsaturatedFrozen

Freezingstarts

Freezingprocedes

Freezing schematization



12

Unsaturatedunfrozen

UnsaturatedFrozen

Freezingstarts

Freezingprocedes

Freezing schematization


-0.10 -0.05 0.00 0.05 0.10

-15

-10

-50

soil suction psi

psi_m=-1m - Tstar= -0.008Temperature [ C]

so

il s

uctio

n p

si [m

] ψfreez

ψw0

T*-10000 -8000 -6000 -4000 -2000 0

0.0

0.1

0.2

0.3

0.4

Psi [mm]

the

ta_

w [

-]


13

Unfrozen water content:

soil water retention curvee.g. Van Genuchten (1980)

Clausius Clapeyron+

!w =pw

"w g

pressure head:

!w(T ) = !w ["w(T )]

Soil Freezing Characteristic curve (SFC)

“freezing=drying” assumption allows to “exploit” the theory of unsaturated soils:


−0.05 −0.04 −0.03 −0.02 −0.01 0.00

0.1

0.2

0.3

0.4

Unfrozen water content

temperature [C]

Thet

a_u

[−]

psi_m −5000

psi_m −1000

psi_m −100

psi_m 0

ice

air

water

...


Soil Freezing Characteristic curve (SFC)

14

depressed melting point

!w = !r + (!s ! !r) ·!

1 +"!"#w0 ! "

Lf

g T0(T ! T !) · H(T ! T !)

#n$"m

ψw0

ψw0

ψw0

ψw0

ψw0



15

Unsaturatedunfrozen

UnsaturatedFrozen

Freezing schematization with SFC

Freezingstarts

Freezingprocedes

θw θw

θwθw



16

Energy conservation

The heat equation below written hides important hypothesis, often tacitly assumed:

Ca!T

!t+ "wcw

JwT

!z=

!

!z

!#

!T

!z

" Harlan (1973)Guymon and Luthin (1974)Fuchs et al. (1978)Zhao et al. (1997)Hansson et al. (2004)Daanen et al. (2007)Watanabe (2008)

apparent heat capacity[J m-3 K -1]

water flux [m s-1]

water density [kg m-3]

temperature[˚C]

mass heat capacity [J kg-1 K -1]

thermal conductivity [W m-1 K -1]



17

Energy conservation

U = hgMg + hwMw + hiMi ! (pwVw + piVi) + µwMphw + µiM

phi

0 assuming equilibrium thermodynamics: µw=µi and Mwph = -Miph



18

Energy conservation

0 assuming freezing=drying


phi




19

Energy conservation



phi

no volume expansion: ρw=ρi

assuming:0no water flux during phase change (closed system)




20

Energy conservation



phi

no volume expansion: ρw=ρi

assuming:0no water flux during phase change (closed system)




21

Energy conservation

!U

!t+ "! • ("G + "J) + Sen = 0

!G = !"T (#w0, T ) · !"T

!J = "w · !Jw(#w0, T ) · [Lf + cw T ]

conduction

advection

• no water flux during phase change (closed system)• freezing=drying• no volume expansion (ρw=ρi)

U = CT · T + !wLf"w

CT := Cg(1! !s) + "wcw!w T + "ici!i

!pfreez ! !wLf

T0(T " T0) closure relation


U = CT · T + !w [Lf ! "w g] #w


22

Energy conservation

!U

!t+ "! • ("G + "J) + Sen = 0

!G = !"T (#w0, T ) · !"T

!J = "w · !Jw(#w0, T ) · [Lf + cw T ]

conduction

advection

• no water flux during phase change (closed system)• freezing=drying• no volume expansion (ρw≠ρi)


!pfreez ! !wLf

T0(T " T0) closure relation


U = CT · T + !w [Lf ! ("w ! "i) g] #w


23

Energy conservation

!U

!t+ "! • ("G + "J) + Sen = 0

!G = !"T (#w0, T ) · !"T

!J = "w · !Jw(#w0, T ) · [Lf + cw T ]

conduction

advection

• no water flux during phase change (closed system)• freezing=drying• no volume expansion (ρw≠ρi)

closure relation


Lf

T0(T ! T0) =

pw

!w! pi

!i Christoffersen and Tulaczyk (2003)



Conclusions

24

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.



25

Thank you!


Education

Thermodynamics of freezing soil