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- 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, 1 Tuesday, May 11, 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? 2 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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 3 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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 4 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 5. References on thermodynamic equilibrium look similar but are actually different... they claim to use the Clausius- Clapeyron relation but... 5 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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 6 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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!!! 8 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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 9 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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 10 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 11. Freezing schematization Unsaturated Freezing unfrozen starts Unsaturated Freezing Frozen procedes 11 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 12. Freezing schematization Unsaturated Freezing unfrozen starts Unsaturated Freezing Frozen procedes 12 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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 13 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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 14 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 15. Freezing schematization with SFC Unsaturated θw θw Freezing unfrozen starts Unsaturated θw θw Freezing Frozen procedes 15 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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] 16 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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 17 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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 18 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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 19 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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 20 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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 21 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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 22 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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 23 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 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. 24 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010
- 25. Thank you! 25 Matteo Dall’Amico et al, EGU 2010 Tuesday, May 11, 2010

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