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 of freezing soils Vienna, 5 may 2010 Tuesday, May 11, 2010
This presentation illustrates the principles of thermodynamics in the freezing soil according to the capillary schematization and the freezing=drying assumption
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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,
assuming:0no water flux during phase change (closed system)
0 assuming equilibrium thermodynamics: µw=µi and Mwph = -Miph
Tuesday, May 11, 2010
Matteo Dall’Amico et al, EGU 2010
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
Tuesday, May 11, 2010
U = CT · T + !w [Lf ! "w g] #w
Matteo Dall’Amico et al, EGU 2010
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)
CT := Cg(1! !s) + "wcw!w T + "ici!i
!pfreez ! !wLf
T0(T " T0) closure relation
Tuesday, May 11, 2010
U = CT · T + !w [Lf ! ("w ! "i) g] #w
Matteo Dall’Amico et al, EGU 2010
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
CT := Cg(1! !s) + "wcw!w T + "ici!i
Lf
T0(T ! T0) =
pw
!w! pi
!i Christoffersen and Tulaczyk (2003)
Tuesday, May 11, 2010
Matteo Dall’Amico et al, EGU 2010
Conclusions
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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.