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◆✉♠❡r✐❝❛❧ s✐♠✉❧❛t✐♦♥ ♦❢ ❛ ❞r♦♣❧❡t ✐❝✐♥❣ ♦♥ ❛ ❝♦❧❞ s✉r❢❛❝❡ ❇✉❝❝✐♥✐ ❙❛✈❡r✐♦ P❡❝❝✐❛♥t✐ ❋r❛♥❝❡s❝♦ ❙✉♣❡r✈✐s♦rs✿ Pr♦❢✳ ❆❧❡ss❛♥❞r♦ ❇♦tt❛r♦ Pr♦❢✳ ❉♦♠✐♥✐q✉❡ ▲❡❣❡♥❞r❡ ▼❛r❝❤ ✸✵✱ ✷✵✶✼ ❯♥✐✈❡rs✐tà ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ●❡♥♦✈❛

❖✉t❧✐♥❡ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ✷✳ P❤②s✐❝❛❧ ♠♦❞❡❧ ❛♥❞ ❡①♣❡r✐♠❡♥ts ✸✳ ◆✉♠❡r✐❝❛❧ ♠❡t❤♦❞ ✹✳ ✶❉ ♣r♦❜❧❡♠ ❛♥❞ ✈❛❧✐❞❛t✐♦♥ ✺✳ ❙✐♠✉❧❛t✐♦♥s ♦❢ t❤❡ ❞r♦♣❧❡t ✻✳ ❈♦♥❝❧✉s✐♦♥s ✶

■♥tr♦❞✉❝t✐♦♥

❚❤❡ ✐❝✐♥❣ ♣r♦❜❧❡♠ ✲ ❊✛❡❝ts ❊✛❡❝ts ❼ ▲✐❢t ✈❛r✐❛t✐♦♥ → st❛❧❧ ❼ ■♥❝r❡❛s❡❞ ❞r❛❣ ❼ ■♥str✉♠❡♥t❛t✐♦♥ ♣r♦❜❧❡♠s → ❆❋✹✹✼ ❼ ■♥❝r❡❛s❡❞ ✇❡✐❣❤t ✷

❚❤❡ ✐❝✐♥❣ ♣r♦❜❧❡♠ ✲ ❈♦✉♥t❡r♠❡❛s✉r❡s ❈♦✉♥t❡r♠❡❛s✉r❡s ❼ ❉❡✲✐❝✐♥❣ ✫ ❛♥t✐✲✐❝✐♥❣ ✢✉✐❞s ❼ ❉❡✲✐❝✐♥❣ ❜♦♦ts ❼ ❊❧❡❝tr✐❝❛❧ r❡s✐st❛♥❝❡s ❼ ❍♦t ❛✐r ❜❧❡❡❞✐♥❣ ❢r♦♠ ❡♥❣✐♥❡s ✸

P❤②s✐❝❛❧ ♠♦❞❡❧ ❛♥❞ ❡①♣❡r✐♠❡♥ts

❋r❡❡③✐♥❣ ❞r♦♣❧❡t ❆✐♠ ♦❢ t❤❡ ✇♦r❦ ❼ ❋r❡❡③✐♥❣ ❢r♦♥t ❡✈♦❧✉t✐♦♥ ❼ ❋✐♥❛❧ s❤❛♣❡ ♦❢ t❤❡ ❞r♦♣❧❡t ❛♥❞ ✐ts ❝❛✉s❡s ✶✳ ❉❡♥s✐t② ✈❛r✐❛t✐♦♥ ✷✳ ▼❛r❛♥❣♦♥✐ ❡✛❡❝t ✹

❙t❡❢❛♥ ♣r♦❜❧❡♠ ✭✶✮ ❼ s♦❧✐❞ ♣❤❛s❡✿ ✵ ≤ x < s ( t ) ∂ ✷ T s ∂ T s ∂ t = α s ∂ x ✷ s ❼ ❧✐q✉✐❞ ♣❤❛s❡✿ s ( t ) < x < ∞ ∂ ✷ T l ∂ T l ∂ t = α l ∂ x ✷ l ❇♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s ❼ ❈♦♥st❛♥t t❡♠♣❡r❛t✉r❡s T ( ✵ , t ) = T w ❛♥❞ T ( x → ∞ , t ) = T i ❼ ■♥t❡r❢❛❝❡ ❝♦♥❞✐t✐♦♥ ❛t x = s ( t ) ✿  � � � �  ∂ T l + λ s ∂ T s = ρ s L ds − λ l  � � ∂ x ∂ x dt � � s + s − � �  s − = T l s + = T m  T s � � ✺

❙t❡❢❛♥ ♣r♦❜❧❡♠ ✭✷✮ ❆ss✉♠♣t✐♦♥s ❼ ❍❡❛t tr❛♥s❢❡r ✐s ❞r✐✈❡♥ ❜② t❤❡ ❆♥❛❧②t✐❝❛❧ s♦❧✉t✐♦♥ s♦❧❡ ❝♦♥❞✉❝t✐♦♥ s = ✷ δ √ α s t ❼ ❚❤❡r♠♦♣❤②s✐❝❛❧ ♣r♦♣❡rt✐❡s ❝♦♥st❛♥t ✇✐t❤ t❡♠♣❡r❛t✉r❡ ✐♥ � � T s ( x , t ) = T wall + ( T m − T wall ) x ✷ √ α s t ❡❛❝❤ ♣❤❛s❡ erf erf ( δ ) ❼ P❤❛s❡ ❝❤❛♥❣❡ t❡♠♣❡r❛t✉r❡ ✜①❡❞ � � T l ( x , t ) = T i − ( T i − T m ) x ✷ √ α l t ❛♥❞ ❦♥♦✇♥ erfc ( δα ) erfc ❼ ❚❤❡ ❡♥t✐r❡ ❞♦♠❛✐♥ ✐♥✐t✐❛❧❧② ❛t T ( x , ✵ ) = T i ✻

❙t❡❢❛♥ ♣r♦❜❧❡♠ ✭✸✮ ❆❞✐♠❡♥s✐♦♥❛❧ ♣❛r❛♠❡t❡rs 25 Ste = ρ s c p , s ( T m − T wall ) 20 ρ s L T 0 =-7°C 15 T i =20°C t=10000s 10 T [°C] ρ l c p , l ( T i − T m ) φ = 5 ρ s c p , s ( T m − T wall ) 0 � α s -5 α = -10 α l 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 x ❊q✉❛t✐♦♥ ❢♦r δ α = δ √ π erf ( δ ) − e − δ ✷ α ✷ e δ ✷ φ erfc ( δα ) Ste ✼

❱❡❧♦❝✐t② ❝❛❧❝✉❧❛t✐♦♥ ❈♦♥t✐♥✉✐t② � � v liq = dH dt = ds ✶ − ρ sol ρ liq V liq + ρ sol V sol = const → dt · ρ liq � �� � r ∂ ✷ T l ∂ T l ∂ T l ●♦✈❡r♥✐♥❣ ❡q✉❛t✐♦♥ ∂ t + v liq ∂ y = α l ∂ x ✷ l ❇❡✐♥❣ t❤❡ ♣❛r❛♠❡t❡r r t❤❡ ✐♥❞✐❝❛t♦r ♦❢ t❤❡ ❡①♣❛♥s✐♦♥✿ � � �� x erfc αδ ✷ δ √ α s t − r T l = T ✵ + ( T m − T ✵ ) erfc [ αδ ( ✶ − r )] e ( r αδ ) ✷ = δ √ π e − δ ✷ e − ( αδ ) ✷ e ✷ r ( αδ ) ✷ erf ( δ ) − φ erfc [ αδ ( ✶ − r )] α Ste ✽

◆✉♠❡r✐❝❛❧ ♠❡t❤♦❞

❏❆❉■▼ ❏❛❞✐♠ ✐s ❛ r❡s❡❛r❝❤ ❝♦❞❡ ❞❡✈❡❧♦♣❡❞ ❜② ❏✳ ▼❛❣♥❛✉❞❡t ❛♥❞ ❉✳ ▲❡❣❡♥❞r❡✬s t❡❛♠ ✐♥ t❤❡ ■♥t❡r❢❛❝❡ ❣r♦✉♣ ❛t ■♥st✐t✉t ❞❡ ▼é❝❛♥✐q✉❡s ❞❡s ❋❧✉✐❞❡s ❞❡ ❚♦✉❧♦✉s❡ ✳ ❚❤❡ ❝♦❞❡ ♣❡r♠✐ts t♦ ❞❡s❝r✐❜❡ ✐♥ ❛♥ ❛❝❝✉r❛t❡ ✇❛② ♣❤②s✐❝❛❧ ♠❡❝❤❛♥✐s♠s ♣r❡s❡♥t ✐♥ ♠✉❧t✐♣❤❛s✐❝ ✢♦✇s✳ ❼ ❱♦❧✉♠❡ ♦❢ ❋❧✉✐❞ ❢♦r♠✉❧❛t✐♦♥ ✐s ❡♠♣❧♦②❡❞ ❼ ❚❤❡r♠❛❧ ❛♥❞ ■♠♠❡rs❡❞ ❇♦✉♥❞❛r② ▼❡t❤♦❞ r♦✉t✐♥❡ ❛r❡ s✉♣♣♦rt❡❞ ■♥ t❤❡ ♣r❡s❡♥t ✇♦r❦ t❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ❝♦✉♣❧❡ t❤❡ t❤r❡❡ ♦❢ t❤❡♠ ❛♥❞✱ ✐♥ ♣❛rt✐❝✉❧❛r✱ t♦ ❞❡✈❡❧♦♣ ❛ t❤❡r♠❛❧ ❜❛s❡❞ ■❇▼ ❢♦r♠✉❧❛t✐♦♥ ✾

❙♦❧✐❞ ❢✉♥❝t✐♦♥ ❢r❛❝t✐♦♥ ❆ s♦❧✐❞ ❢✉♥❝t✐♦♥ ❢r❛❝t✐♦♥ ❤❛s ❜❡❡♥ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✱ r❡♣r❡s❡♥t✐♥❣ t❤❡ ❛♠♦✉♥t ♦❢ ✐❝❡ ❢♦r ❡❛❝❤ ❝❡❧❧✿ T max − T α ibm , lin = τ · T max − T min � � cos π · ( T − T min ) α ibm , cos = ✵ . ✺ · τ · + ✶ T max − T min 1 ibm,linear 0.8 ibm,cos 0.6 ibm 0.4 0.2 0 -10 -8 -6 -4 -2 0 2 4 6 8 10 T [°C] φ = ( ✶ − τ ) φ air + τ [( ✶ − α ibm ) φ liq + α ibm φ sol ] ✶✵

■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ✈❡❧♦❝✐t② ❱❡❧♦❝✐t② ❝❛❧❝✉❧❛t✐♦♥ ❱❡❧♦❝✐t② ✐♠♣♦s✐t✐♦♥ � � ❯ − ❯ ∗ ✶ − ρ ice ❢ = α ibm ✈ liq = ❱ front ∆ t ρ water � ❙❝❛❧❛r tr❛♥s♣♦rt ❡q✉❛t✐♦♥✿ U = ✵ α ibm ≥ ✵ . ✾✺ U = v liq ✵ < α ibm < ✵ . ✾✺ ∂α ibm + ❱ front · ∇ α ibm = ✵ ❇❡✐♥❣ ❯ ∗ ❛ ♣r❡❞✐❝t♦r ✈❡❧♦❝✐t② ✇✐t❤♦✉t ∂ t ❝♦♥s✐❞❡r✐♥❣ t❤❡ ✐♠♠❡rs❡❞ ♦❜❥❡❝t ❱ front · ♥ = − ∂α ibm ∂ T ∂ T ∂ t Pr❡ss✉r❡ ❝♦rr❡❝t✐♦♥ ||∇ α ibm || ❏❛❞✐♠✬s ❙■▼P▲❊ ❛❧❣♦r✐t❤♠ ✐s ∂α ibm ♠♦❞✐✜❡❞ ✐♥ ♦r❞❡r t♦ t❛❦❡ ❛❝❝♦✉♥t ♦❢ ∂ x i n i = ||∇ α ibm || t❤❡ ❝❛❧❝✉❧❛t❡❞ ✈❡❧♦❝✐t✐❡s ✶✶

❚❤❡ ❧❛t❡♥t ❤❡❛t ❝♦♠♣✉t❛t✐♦♥ � H = c p , l T ▲✐q✉✐❞ H = c p , s T + L ❙♦❧✐❞ ∂ρ H = ∇ · ( k ∇ T ) ∂ t ❚❤❡ ❛♣♣❛r❡♥t ❤❡❛t ❝❛♣❛❝✐t② ♠❡t❤♦❞ ❚❤❡ s♦✉r❝❡ t❡r♠ ♠❡t❤♦❞ c app = dH � � dT ∂ T ∂ t = ∂ k ∂ T ρ c p − S ∂ x ∂ x c app = c p + Ld α IBM dT [ L + T ( c p , s − c p , l )] S = ∂α IBM ∂ T ∂ T ∂ t c p ∂ T ρ c app ∂ t = ∇ · ( k ∇ T ) ✶✷

✶❉ ♣r♦❜❧❡♠ ❛♥❞ ✈❛❧✐❞❛t✐♦♥

■❇▼ ❢✉♥❝t✐♦♥s err T , max err T , avg err int , max err int , avg ▲✐♥❡❛r [ − ✷ , ✷ ] ✻✳✶✪ ✺✳✼✪ ✷✼✪ ✶✼✪ ▲✐♥❡❛r [ − ✶ , ✶ ] ✹✳✸✪ ✹✳✶✪ ✶✹✳✷✪ ✾✳✷✪ ▲✐♥❡❛r [ ✵ , ✶ ] ✺✳✷✪ ✹✳✷✪ ✹✳✼✪ ✷✳✷✪ ▲✐♥❡❛r [ ✵ , ✵ . ✶ ] ✷✳✽✪ ✷✪ ✸✳✺✪ ✶✪ ❈♦s✐♥❡ [ − ✶ , ✶ ] ✺✳✷✪ ✹✳✸✪ ✶✷✳✷✪ ✼✳✻✪ ❈♦s✐♥❡ [ ✵ , ✶ ] ✺✳✸✪ ✹✳✶✪ ✹✳✽✪ ✷✳✸✪ ❼ ❋♦r ♥❛rr♦✇❡r s♦❧✐❞✐✜❝❛t✐♦♥ r❛♥❣❡s✱ t❤❡ r❡s✉❧ts ❛r❡ ♠♦r❡ ❛❝❝✉r❛t❡ ❼ ❚❤❡ s♦❧✐❞✐✜❝❛t✐♦♥ ❢r♦♥t ✐s ✇❡❧❧ ❧♦❝❛t❡❞ ✐❢ t❤❡ ✐♥❢❡r✐♦r ❧✐♠✐t ❝♦✐♥❝✐❞❡s t♦ ✵ ◦ C ✶✸

■❇▼ ❢✉♥❝t✐♦♥s✱ ✜❣✉r❡s ❋r❡❡③✐♥❣ ❢r♦♥t✱ ❧✐♥ [ ✵ , ✵ . ✶ ] ❋r❡❡③✐♥❣ ❢r♦♥t✱ ❝♦s [ − ✶ , ✶ ] ✶✹