Modelling and simulation of ice/snow melting

Modelling and simulation of ice/snow 

melting 

Sabrina Wandl - University of Linz 

Tuomo Mäki-Marttunen - Tampere UT 

Sigmund Vestergaard - TU Danmark 

Patrick Kürschner - TU Chemnitz 

Trond Kvarnsdal - NTNU Trondheim 

ECMI Modelling week 2008 

Modelling and 

simulation of ice/snow 


Sabrina Wandl - 

University of Linz 

Tuomo 

Mäki-Marttunen - 

Tampere UT 

Sigmund Vestergaard - 

TU Danmark 

Patrick Kürschner - 

TU Chemnitz 

Trond Kvarnsdal - 

NTNU Trondheim 

Introduction 

Modelling 

Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues

Outline 

Introduction 


Melting point depression 

Solution 

Exact 

Numerical 

Further issues 






Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues

Introduction 

Problem Description 

◮ find model for melting processes 

◮ consideration of effects caused by presence of salt 

→ melting point depression 

◮ numerical solution 






Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues







Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues


heat equations 

◮ liquid phase: 

◮ solid phase: 

∂TL 

∂t 

= kL 

ρLcL 

TL(0, t) = TA 

∂TS 

∂t 

= kS 

ρScS 

TS(l, t) = TG 

◮ thermophysical constants: 

∂2TL , 0 ≤ x ≤ h(t) 

∂x 2 

∂2TS , h(t) ≤ x ≤ l 

∂x 2 

kS, kL − thermal conductivities 

cS, cL − specific heat 

ρS = ρL − densities 






Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues


interface movement 

◮ TS(h(t), t) = TL(h(t), t) = TM (melting point) 

◮ domain Ω = [0, l] separated by interface into 

[0, h(t)) and (h(t), l] 

◮ one-dimensional Stefan condition: 

∂TS 

∂TL 

kS (h(t), t) − kL (h(t), t) 

� 

∂x 

�� 

∂x 

� 

heat flux difference 

h(0) = 0 slap initially frozen 

L - latent heat (e.g. water: 334 J/g) 

= ρL dh 

dt 

�� 

velocity 






Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues


Two-phase Stefan problem 

� Tt = αLTxx : 0 < x < h(t), αL := kL 

ρcL , 

Heat-Equations: 

Tt = αSTxx : h(t) < x < l, αS := kS 

ρcS . 

� 

Interface conditions T (h(t), t) = TM, 

for t > 0 : ρLh ′ (t) 

� 

= kSTx(h(t), t)−kLTx(h(t), t). 

h(0) 

Initial conditions: 

T (x, 0) 

� 

= 0 (material initially solid), 

= Tinit < TM : 0 ≤ x ≤ l. 

Boundary conditions T (0, t) = TA > TM, 

for t > 0 : T (l, t) = TG < TM. 






Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues


Simplification: solid phase at uniform temperature Tinit = TM, 

only liquid phase is active, Ω = [0, ∞) 

=⇒ one phase Stefan problem 

Find T (x, t) and h(t) such that 

Heat equation: Tt = αLTxx, 0 < x < h(t), t > 0, 

Interface conditions 

for t > 0 : 

Initial condition: h(0) = 0. 

� 

T (h(t), t) = TM, 

h ′ (t) = −βTx(h(t), t), β := kL 

ρL 

Boundary condition: T (0, t) = TA(t) = TA > TM : t > 0. 

similar processes: solidification, flows through porous media, 

combustion, reactions 






Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues


(a) (b) 

◮ without salt: dynamic equilibrium (a) at melting point 

◮ with salt (i.e. NaCl): disruption of equilibrium (b) 

◮ foreign molecules (i.e. Na + , Cl − ) dissolve in the water 

◮ the chemical potential of the solvent is decreased by dilution 

◮ equilibrium between solid and liquid phase is established at 

another depressed melting point 






Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues


◮ Raoul’s law: ∆TM = KM · mB 

melting point depression: 

∆TM = T M(pure solvent) − T M(solution) 

KM - the cryoscopic constant 

mB - the molality of the solution (no. of moles of solute per kg 

solvent) 

◮ molality mB = msolute · i, 

i- van’t Hoff factor (number of individual particles in solution) 

⇒ colligative property 

◮ sugar: i = 1 

◮ NaCl: i = 2 (Na + and Cl − ions) 

◮ CaCl2: i = 3 (Ca 2+ and 2 Cl − ions) 






Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues

Analytical Solution 

If the initial temperature distribution is 

⎧ 

⎪⎨ 

T (x, 0)= 

⎪⎩ TA−(TA−TM) 

� 

x 

erf 

2 √ � 

αt0 

erf(λ) , 0≤x ≤h(0) 

TM, x > h(0) 

with: 

λ - the solution of equation λeλ2 =⇒ solution of the Stefan problem: 

h(t) = 2λ � α(t + t0) 

erf 

T (x, t) = TA − (TA − TM) 

erf(λ) = CL(TA−TM ) 

√ πL 

� 

x 

2 √ α(t+t0) 

erf(λ) 

and t0 = h(0)2 

4λ 2 α 

� 






Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues

Numerical Solution 

Numerical solution of Stefan problem with FDM: 

For each time step: 

1. Update the temperature distribution by Tt = αTxx using 

implicit FDM: 

T (x −∆x, t +∆t) 

T (x, t +∆t) = T (x, t)+α ∆t 

− α 

∆x 2 

2T (x, t +∆t)−T (x +∆x, t +∆t) 

∆x 2 

∆t 

2. Update the boundary state by ht = βTx(h(t), t) using explicit 

FDM: 

h(t + ∆t) = β 

3. new melting point 

T (h(t), t) − T (h(t) − ∆x, t) 

∆t 

∆x 






Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues


1. Temperature distribution 

Using matrix notation 

⎡ ⎤ 

TA 

⎢ 

T2(t) ⎥ 

⎢ 

. ⎥ 

. ⎥ 

⎢ . ⎥ 

⎢ ⎥ 

T (t +∆t)= ⎢Tn(t) 

⎥+ 

⎢ ⎥ 

⎢ TF ⎥ 

⎢ ⎥ 

⎢ . 

⎣ . 

⎥ 

. ⎦ 

TF 

∆t 

h2 ⎡ 

0 

⎢ 

1 

⎢ 

⎣0 

0 

� 

−2 

1 

. . . 

1 

. .. 

−2 

. . . 

�� 

=A 

0 

1 

. .. 

⎡ 

⎤ 

⎤ T1(t + ∆t) 

0 

⎢ 

0⎥ 

⎢ 

. ⎥ 

. ⎥ 

⎥ ⎢ . ⎥ 

⎥ ⎢ 

⎥ 

⎥ ⎢Tn−1(t 

+ ∆t) ⎥ 

⎥ ⎢ 

⎥ 

⎥ ⎢ Tn(t + ∆t) ⎥ 

⎥ ⎢ 

⎥ 

⎥ ⎢Tn+1(t 

+ ∆t) ⎥ 

⎥ ⎢ 

⎥ 

0⎦ 

⎢ . 

⎣ . 

⎥ 

. ⎦ 

0 

� TN+1(t + ∆t) 

leads to update step algorithm: 

T (t +∆t)=(IN+1−A) −1 [TA,T2(t),· · · ,Tn(t),TF ,· · · ,TF ] T 






Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues


2. Interface movement 

Update step for interface movement 

h(t + ∆t) = h(t) + β Tn+1(t) − Tn(t) 

∆t 

∆x 

3. Melting point depression 

Assumption: constant amount of substance of solute per unit area 

nA, V = h(t) · A 

TM(t) = T M(pure) − KMmb = . . . = T M(pure) − KMi nAAsolution 

ρVsolution(t) 

= T M(pure) − KMi nA 

ρh(t) 






Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues


Example 






Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues

Further Issues 

Possible improvements 

◮ extend model to more dimensions and other geometries 

◮ consider more than one phase 

◮ supercooling effects 

◮ ρL �= ρS (ρL < ρS ⇒ void formation) 

◮ variable thermophysical properties, i.e. c(t), k(t), ρ(t) 

◮ diffusion model for salt concentration 

◮ include other strategies for solution 

◮ FEM 

◮ pertubation methods 






Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues

Further Issues 

Possible improvements 

◮ extend model to more dimensions and other geometries 

◮ consider more than one phase 

◮ supercooling effects 

◮ ρL �= ρS (ρL < ρS ⇒ void formation) 

◮ variable thermophysical properties, i.e. c(t), k(t), ρ(t) 

◮ diffusion model for salt concentration 

◮ include other strategies for solution 

◮ FEM 

◮ pertubation methods 

Thanks for your Attention ! 






Tuomo 


Tampere UT 


TU Danmark 


TU Chemnitz 



Introduction 


Melting point 

depression 

Solution 

Exact 

Numerical 

Further issues

Modelling and simulation of ice/snow melting

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