Master thesis presentation by Niccolò Tubini
-
Upload
riccardo-rigon -
Category
Education
-
view
28 -
download
2
Transcript of Master thesis presentation by Niccolò Tubini
![Page 1: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/1.jpg)
Department of Civil, Environmental and Mechanical Engineering
Master of Science inEnvironmental and Land Engineering
THEORETICAL PROGRESS INFREEZING – THAWING PROCESSES STUDY
Supervisor StudentProf. Riccardo Rigon Niccolò Tubini
Co-supervisorsProf. Stephan GruberDr. Francesco Serafin
![Page 2: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/2.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
This work is licensed under a CreativeCommons “Attribution-ShareAlike 4.0International” license.
Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 3: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/3.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
What is the purpose?
The aim of my Master’s thesis is to develop a newinterpretation of modeling freezing soils.
Niccolò Tubini Theoretical progress in freezing – thawing processes study 2 / 34
![Page 4: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/4.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Why studing the influence of coupled heat and water flowin soils?
Freezing – thawing processes entail a hugeexchange of heat;
To simulate more realistic soil temperature(Luo et al., 2003).
Studies have shown that proper frozen soilschemes help improve climate model simulation(Viterbo et al., 1999 and Smirnova et al., 2000).
Niccolò Tubini Theoretical progress in freezing – thawing processes study 3 / 34
![Page 5: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/5.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Why studing the influence of coupled heat and water flowin soils?
Freezing – thawing processes entail a hugeexchange of heat;
To simulate more realistic soil temperature(Luo et al., 2003).
Studies have shown that proper frozen soilschemes help improve climate model simulation(Viterbo et al., 1999 and Smirnova et al., 2000).
Niccolò Tubini Theoretical progress in freezing – thawing processes study 3 / 34
![Page 6: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/6.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Why studing the influence of coupled heat and water flowin soils?
Freezing – thawing processes entail a hugeexchange of heat;To simulate more realistic soil temperature(Luo et al., 2003).
Studies have shown that proper frozen soilschemes help improve climate model simulation(Viterbo et al., 1999 and Smirnova et al., 2000).
Niccolò Tubini Theoretical progress in freezing – thawing processes study 3 / 34
![Page 7: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/7.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soils
Some definitions
Air gas
Liquid water
Soil particle
Va
Vw
Vs
Vc
Niccolò Tubini Theoretical progress in freezing – thawing processes study 4 / 34
![Page 8: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/8.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soils
Some definitions
Soil porosity
φ := Vs
Vc
Water content
θ := Vw
Vc
Niccolò Tubini Theoretical progress in freezing – thawing processes study 5 / 34
![Page 9: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/9.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soils
Some definitions
Assuming the rigid soil scheme
θs := φ
0 < θr ≤ θ ≤ θs < 1
Niccolò Tubini Theoretical progress in freezing – thawing processes study 6 / 34
![Page 10: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/10.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soils
Young – Laplace equation
r γaw
α pw = pa −2γaw cosα
r
Niccolò Tubini Theoretical progress in freezing – thawing processes study 7 / 34
![Page 11: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/11.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soils
Young – Laplace equation
pa ← 0
Let us define suction as
ψ := pw
gρw
Niccolò Tubini Theoretical progress in freezing – thawing processes study 8 / 34
![Page 12: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/12.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Mualem’s assumption
Wetting and drying processes are assumed to beselective processes.
Niccolò Tubini Theoretical progress in freezing – thawing processes study 9 / 34
![Page 13: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/13.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Water – retention – hydraulic – conductivity models
Dealing with unsaturated soils requires thedefinition of the relationship between
θ–ψ and K–ψ
Niccolò Tubini Theoretical progress in freezing – thawing processes study 10 / 34
![Page 14: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/14.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Empirical curve-fitting models
Parameters of these models have been related tothe soil texture and other soil properties
Despite their usfulness they do not emphasize thephysical significance of their empirical parameters
Niccolò Tubini Theoretical progress in freezing – thawing processes study 11 / 34
![Page 15: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/15.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Empirical curve-fitting models
Parameters of these models have been related tothe soil texture and other soil properties
Despite their usfulness they do not emphasize thephysical significance of their empirical parameters
Niccolò Tubini Theoretical progress in freezing – thawing processes study 11 / 34
![Page 16: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/16.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Lognormal distribution model (Kosugi, 1996)
The idea is to derive the water retention curve fromthe pore-size distribution:
f (r) := dθdr
Niccolò Tubini Theoretical progress in freezing – thawing processes study 12 / 34
![Page 17: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/17.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Lognormal distribution model (Kosugi, 1996)
r0
50
100
150
f(r)
R
Water
f (r) = θs − θr√2π σr
exp
−[ln( r
rm
)]22σ2
θ(R) = θr +∫ R
0f (r)dr
Niccolò Tubini Theoretical progress in freezing – thawing processes study 13 / 34
![Page 18: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/18.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Lognormal distribution model (Kosugi, 1996)
Young-Laplace equation allows to transform thepore-size distribution into the capillary pressure
distribution function
g(ψ) = f (r) drdψ
Niccolò Tubini Theoretical progress in freezing – thawing processes study 14 / 34
![Page 19: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/19.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Unsaturated soil hydraulic properties
Lognormal distribution model (Kosugi, 1996)
θ(Ψ) = θr +∫ Ψ
−∞g(ψ)dψ
Ψ = −2γaw cosαg ρw R
Niccolò Tubini Theoretical progress in freezing – thawing processes study 15 / 34
![Page 20: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/20.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Some definitions
Air gas
Ice
Liquid water
Particle soil
Va
Vi
Vw
Vs
Vc
Niccolò Tubini Theoretical progress in freezing – thawing processes study 16 / 34
![Page 21: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/21.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Some definitions
Liquid water content
θw := Vw
Vc
Ice content
θi := Vi
Vc
Niccolò Tubini Theoretical progress in freezing – thawing processes study 17 / 34
![Page 22: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/22.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Some definitions
Total water content
θ := θw + θi
0 < θr ≤ θ ≤ θs < 1
Niccolò Tubini Theoretical progress in freezing – thawing processes study 18 / 34
![Page 23: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/23.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Model assumptions
Model assumptions
rigid soil scheme
freezing = drying (Miller, 1965; Spaans and Baker, 1996)
the phase change is assumed to occur at thethermodynamic equilibrium
Niccolò Tubini Theoretical progress in freezing – thawing processes study 19 / 34
![Page 24: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/24.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Model assumptions
Model assumptions
rigid soil scheme
freezing = drying (Miller, 1965; Spaans and Baker, 1996)
the phase change is assumed to occur at thethermodynamic equilibrium
Niccolò Tubini Theoretical progress in freezing – thawing processes study 19 / 34
![Page 25: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/25.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Model assumptions
Model assumptions
rigid soil scheme
freezing = drying (Miller, 1965; Spaans and Baker, 1996)
the phase change is assumed to occur at thethermodynamic equilibrium
Niccolò Tubini Theoretical progress in freezing – thawing processes study 19 / 34
![Page 26: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/26.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Freezing point depression
Gibbs-Thomson equation (Acker et al., 2001)
Tm − T ∗ = 2 γaw Tm cosαρw ` r + πw Tm
ρw `
Capillary effectDissolved solutes
Niccolò Tubini Theoretical progress in freezing – thawing processes study 20 / 34
![Page 27: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/27.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Freezing point depression
Gibbs-Thomson equation (Acker et al., 2001)
The ice-water interface occurs at:
r̂(T ) := −2 γaw Tm cosαρw `(T − Tm) for T < Tm∗
Niccolò Tubini Theoretical progress in freezing – thawing processes study 21 / 34
![Page 28: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/28.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Water and ice content
Let us define
r ∗ :=R if r̂ ≥ R or T ≥ Tm
r̂ otherwise
∂r ∗∂t :=
∂R∂t if r̂ ≥ R or T ≥ Tm
∂ r̂∂t otherwise
Niccolò Tubini Theoretical progress in freezing – thawing processes study 22 / 34
![Page 29: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/29.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Water and ice content
r0
50
100
150
f(r)
Rr̂ = r ∗
WaterIce
θw = θr +∫ r∗
0f (r)dr
θi =∫ R
r∗f (r)dr
Niccolò Tubini Theoretical progress in freezing – thawing processes study 23 / 34
![Page 30: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/30.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Water and ice content
The phase change rate
θi =∫ R
r∗f (r)dr
∂θi
∂t = ∂R∂t f (R)− ∂r ∗
∂t f (r ∗)
Niccolò Tubini Theoretical progress in freezing – thawing processes study 24 / 34
![Page 31: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/31.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Water and ice content
The phase change rate
r0
20
40
60
80
100
120f(
r)
R(t) R(t + δt)r̂
WaterIce at time tIce at time t + δt
Niccolò Tubini Theoretical progress in freezing – thawing processes study 25 / 34
![Page 32: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/32.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Water and ice content
The phase change rate
r0
20
40
60
80
100
120f(
r)
Rr̂(t)r̂(t + δt)
WaterIce formed in δtIce at time t
Niccolò Tubini Theoretical progress in freezing – thawing processes study 26 / 34
![Page 33: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/33.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Water and ice content
The phase change rate
∂θi
∂t = ∂Ψ∂t g(Ψ)− ∂ψ∗
∂t g(ψ∗)
Niccolò Tubini Theoretical progress in freezing – thawing processes study 27 / 34
![Page 34: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/34.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Mass conservation equation
θ
J ET
~Jw
∂
∂t (ρwθw + ρiθi) = −ρw∇ · ~Jw
Water flux:~Jw = −K (ψ∗) ~∇(ψ∗ + z)
Niccolò Tubini Theoretical progress in freezing – thawing processes study 28 / 34
![Page 35: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/35.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Mass conservation equation
θ
J ET
~Jw
∂
∂t (ρwθw + ρiθi) = −ρw∇ · ~Jw
Water flux:~Jw = −K (ψ∗) ~∇(ψ∗ + z)
Niccolò Tubini Theoretical progress in freezing – thawing processes study 28 / 34
![Page 36: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/36.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Mass conservation equation
Setting ρw = ρi
∂θ
∂t = ∂Ψ∂t g(Ψ) = ∇ · [K (ψ∗)~∇(ψ∗ + z)]
∂θi
∂t = ∂Ψ∂t g(Ψ)− ∂ψ∗
∂t g(ψ∗)
∂θw
∂t = ∂θ
∂t −∂θi
∂tNiccolò Tubini Theoretical progress in freezing – thawing processes study 29 / 34
![Page 37: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/37.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Energy conservation equation
ε
J
HRn
ET
~Jw~Jg
∂ε
∂t = −∇ · (~Jw + ~Jg)
Advective flux:~Jw = ~Jw ρw [` + cw (T − Tm)]
Heat conduction:~Jg = −λ~∇T
Niccolò Tubini Theoretical progress in freezing – thawing processes study 30 / 34
![Page 38: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/38.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Energy conservation equation
ε
J
HRn
ET
~Jw~Jg
∂ε
∂t = −∇ · (~Jw + ~Jg)
Advective flux:~Jw = ~Jw ρw [` + cw (T − Tm)]
Heat conduction:~Jg = −λ~∇T
Niccolò Tubini Theoretical progress in freezing – thawing processes study 30 / 34
![Page 39: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/39.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Energy conservation equation
Setting ρw = ρi
CT∂T∂t − ρi`
(∂Ψ∂t g(Ψ)− ∂ψ∗
∂t g(ψ∗))
−ρi(cw − ci)(T − Tm)(∂Ψ∂t g(Ψ)− ∂ψ∗
∂t g(ψ∗))
+ρicw~Jw · ~∇T + ρigz∇ · ~Jw −∇ · ~Jg = 0
Niccolò Tubini Theoretical progress in freezing – thawing processes study 31 / 34
![Page 40: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/40.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Energy conservation equation: if ice occurs
ψ∗ := ψ̂∂ψ∗
∂t := `
g Tm
∂T∂t
Cph∂T∂t − ρi [` + (cw − ci)(T − Tm)]∂Ψ
∂t g(Ψ)
+ ρicw~Jw · ~∇T + ρigz∇ · ~Jw −∇ · ~Jg = 0
Niccolò Tubini Theoretical progress in freezing – thawing processes study 32 / 34
![Page 41: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/41.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
The apparent heat capacity
Cph := CT + ρi [` + (cw − ci)(T − Tm)] `
g Tm
CT := ρscs(1− θs) + ρiciθi + ρwcwθw
Niccolò Tubini Theoretical progress in freezing – thawing processes study 33 / 34
![Page 42: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/42.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
The apparent heat capacity
Cph := CT + ρi [` + (cw − ci)(T − Tm)] `
g Tm
CT := ρscs(1− θs) + ρiciθi + ρwcwθw
Niccolò Tubini Theoretical progress in freezing – thawing processes study 33 / 34
![Page 43: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/43.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
To take home
Freezing=drying and rigid soil schemeassumptions are useful when freezing-inducedmechanical deformations are not considered;
Freezing/thawing processes do not occur at thethermodynamic equilibrium (Kurylyk, 2013).
Kosugi retention model has the benefit to bestraightforward extended to freezing soils caseby making use of Gibbs – Thomson equation;
Niccolò Tubini Theoretical progress in freezing – thawing processes study 34 / 34
![Page 44: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/44.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
To take home
Freezing=drying and rigid soil schemeassumptions are useful when freezing-inducedmechanical deformations are not considered;
Freezing/thawing processes do not occur at thethermodynamic equilibrium (Kurylyk, 2013).
Kosugi retention model has the benefit to bestraightforward extended to freezing soils caseby making use of Gibbs – Thomson equation;
Niccolò Tubini Theoretical progress in freezing – thawing processes study 34 / 34
![Page 45: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/45.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
To take home
Freezing=drying and rigid soil schemeassumptions are useful when freezing-inducedmechanical deformations are not considered;Freezing/thawing processes do not occur at thethermodynamic equilibrium (Kurylyk, 2013).
Kosugi retention model has the benefit to bestraightforward extended to freezing soils caseby making use of Gibbs – Thomson equation;
Niccolò Tubini Theoretical progress in freezing – thawing processes study 34 / 34
![Page 46: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/46.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
To take home
This formulation allows to take into account ofdissolved solutes;
It is possible to solve the mass and energyequation in a decoupled way;
Niccolò Tubini Theoretical progress in freezing – thawing processes study 35 / 34
![Page 47: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/47.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
To take home
This formulation allows to take into account ofdissolved solutes;
It is possible to solve the mass and energyequation in a decoupled way;
Niccolò Tubini Theoretical progress in freezing – thawing processes study 35 / 34
![Page 48: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/48.jpg)
Ottaw
aRiver,17
thDec
2016
![Page 49: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/49.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
References
I K. Kosugi, Lognormal distribution model for unsaturatedsoil hydraulic properties, Water Resources Research, vol. 32,no. 9, pp. 2697–2703, 1996.
I J. T. Acker et al., Intercellular ice propagation:experimental evidence for ice growth through membranepores, Biophysical journal, vol. 81, no. 3, pp. 1389–1397,2001.
I M. Dall’Amico et al., A robust and energy-conservingmodel of freezing variably-saturated soil, The Cryosphere,vol. 5, no. 2, p. 469, 2011.
I M. Dall’Amico, Coupled water and heat transfer inpermafrost modeling, Ph.D. dissertation, University ofTrento, 2010.Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 50: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/50.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
References
I E. J. Spaans and J. M. Baker, The soil freezingcharacteristic: Its measurement and similarity to the soilmoisture characteristic, Soil Science Society of AmericaJournal, vol. 60, no. 1, pp. 13–19, 1996.
I R. D. Miller, Phase equilibria and soil freezing, vol. 287, pp.193–197, 1965.
I B. L. Kurylyk and K. Watanabe, The mathematicalrepresentation of freezing and thawing processes invariably-saturated, non-deformable soils, Advances in WaterResources, vol. 60, pp. 160–177, 2013.
Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 51: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/51.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
References
I L. Luo et al., Effects of frozen soil on soil temperature,spring infiltration, and runoff: Results from the PILPS 2 (d)experiment at Valdai, Russia, Journal of Hydrometeorology,vol. 4, no. 2, pp. 334–351, 2003.
I T. G. Smirnova, J. M. Brown, S. G. Benjamin, and D. Kim,Parameterization of cold-season processes in the mapsland-surface scheme, Journal of Geophysical Research:Atmospheres, vol. 105, no. D3, pp. 4077– 4086, 2000.
I P. Viterbo, A. Beljaars, J.-F. Mahfouf, and J. Teixeira, Therepresentation of soil moisture freezing and its impact onthe stable boundary layer, Quarterly Journal of the RoyalMeteorological Society, vol. 125, no. 559, pp. 2401–2426,1999.Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 52: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/52.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Freezing=drying assumptionDall’A
mico,
2010
pa
pw (R)
R
r
Air-water interfacepw (R) = pa −
2 γaw cosαR
Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 53: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/53.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Freezing=drying assumptionDall’A
mico,
2010
pa
pi
pw (r)
R
r
Air-ice interfacepi = pa −
2 γai cosαR
Ice-water interfacepw (r) = pi −
2 γiw cosαr
Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 54: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/54.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Freezing=drying assumptionDall’A
mico,
2010
pa
pi ≡ pa
pw (r)
R
r
Air-water interfacepw (r) = pa −
2 γaw cosαr
Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 55: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/55.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
r0
20
40
60
80
100
120f(
r)
R = r ∗ r̂
Water
Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 56: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/56.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
The phase change rate
Thanks to the Young-Laplace equation
ψ∗ :=
Ψ if r̂ ≥ 2 γaw cosα
ρw g Ψ or T ≥ Tm
ψ̂ = ψ(r̂) otherwise
Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 57: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/57.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
The phase change rate
Thanks to the Young-Laplace equation
∂ψ∗
∂t :=
∂Ψ∂t if r̂ ≥ 2γaw cosα
ρwgΨ or T ≥ Tm
∂ψ̂
∂t otherwise
Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 58: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/58.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Comparison with Dall’Amico model (Dall’Amico et al., 2011)
By making use of Clausius – Clapeyron equation:
dTdpw
= Tρw `
T ∗ = Tm + g Tm
`ψw0
ψ(T ) = ψw0 + `
g T ∗ (T − T ∗)H(T ∗ − T )
Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 59: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/59.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Comparison with Dall’Amico model (Dall’Amico et al., 2011)
By making use of Clausius – Clapeyron equation:
dTdpw
= Tρw `
T ∗ = Tm + g Tm
`ψw0
ψ(T ) = ψw0 + `
g T ∗ (T − T ∗)H(T ∗ − T )
Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 60: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/60.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Mass conservation equation: if there is no ice
ψ∗ := Ψ ∂ψ∗
∂t := ∂Ψ∂t
∂θ
∂t = ∂Ψ∂t g(Ψ) = ∇ · [K (Ψ)~∇(Ψ + z)]
∂θi
∂t =���
������
�����:0
∂Ψ∂t g(Ψ)− ∂ψ∗
∂t g(ψ∗)
∂θw
∂t = ∂θ
∂tNiccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 61: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/61.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Mass conservation equation: if ice occurs
ψ∗ := ψ̂∂ψ∗
∂t := `
g Tm
∂T∂t
∂θ
∂t = ∂ψ̂
∂t g(Ψ) = ∇ · [K (ψ̂)~∇(ψ̂ + z)]
∂θi
∂t = ∂Ψ∂t g(Ψ)− ∂ψ̂
∂t g(ψ̂)
∂θw
∂t = ∂θ
∂t −∂θi
∂tNiccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 62: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/62.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Energy conservation equation: if there is no ice
ψ∗ := Ψ ∂ψ∗
∂t := ∂Ψ∂t
CT∂T∂t − ρi l
������
������
����:0(
∂Ψ∂t g(Ψ)− ∂ψ∗
∂t g(ψ∗))
− ρi(cw − ci)(T − Tm)���
������
������
�:0(∂Ψ∂t g(Ψ)− ∂ψ∗
∂t g(ψ∗))
+ ρicw~Jw · ~∇T + ρigz∇ · ~Jw −∇ · ~Jg = 0Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 63: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/63.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Energy conservation equation: if there is no ice
CT∂T∂t + ρwcw~Jw · ~∇T + ρwgz∇ · ~Jw −∇ · ~Jg = 0
CT := ρscs(1− θs) + ρiciθi + ρwcwθw
Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 64: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/64.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Energy conservation equation: if there is no ice
CT∂T∂t + ρwcw~Jw · ~∇T + ρwgz∇ · ~Jw −∇ · ~Jg = 0
CT := ρscs(1− θs) + ρiciθi + ρwcwθw
Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 65: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/65.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Numerical scheme for unfrozen soils
The mass conservation equation ⇒ Nested Newtonmethod (Casulli and Zanolli, 2010).
The energy consevation equation ⇒ Implicit upwindmethod
Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 66: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/66.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Numerical scheme for frozen soils
The mass conservation equation becomes∂θ
∂t = ∇ ·(K (ψ̂) ~∇(ψ̂ + z)
)
Nested Newton method (Casulli and Zanolli, 2010).should be extended for equations of two variables
The energy consevation equation ⇒ Implicit upwindmethod
Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 67: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/67.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Numerical scheme for frozen soils
The mass conservation equation becomes∂θ
∂t = ∇ ·(K (ψ̂) ~∇(ψ̂ + z)
)
Nested Newton method (Casulli and Zanolli, 2010).should be extended for equations of two variables
The energy consevation equation ⇒ Implicit upwindmethod
Niccolò Tubini Theoretical progress in freezing – thawing processes study
![Page 68: Master thesis presentation by Niccolò Tubini](https://reader031.fdocuments.in/reader031/viewer/2022022413/58ed43231a28ab636b8b457d/html5/thumbnails/68.jpg)
Introduction Water in soils Freezing soils Mass conservation Energy conservation Conclusions
Numerical scheme for frozen soils
The mass conservation equation becomes∂θ
∂t = ∇ ·(K (ψ̂) ~∇(ψ̂ + z)
)
Nested Newton method (Casulli and Zanolli, 2010).should be extended for equations of two variables
The energy consevation equation ⇒ Implicit upwindmethod
Niccolò Tubini Theoretical progress in freezing – thawing processes study