phase change materials - comsol multiphysics...pcm donato rubinetti appendix a - modeling functions...

PCM

DonatoRubinetti

Introduction

PhysicalModel

NumericalModel

ApplicationExample

Final Remarks

1

Phase Change MaterialsModeling Approach to Facilitate Thermal Energy Management in Buildings

Donato Rubinetti 1 Daniel A. Weiss 1 Arnab Chaudhuri 2 Dimitrios Kraniotis 2

1Institute of Thermal and Fluid EngineeringUniversity of Applied Sciences and Arts Northwestern Switzerland

2Department of Civil Engineering and Energy TechnologyOslo Metropolitan University, Norway

27.09.2018

PCM

DonatoRubinetti

Introduction

Application

Modeling

PhysicalModel

NumericalModel

ApplicationExample

Final Remarks

2

IntroductionWhat are PCMs and what are their application areas?

• Materials with a characteristically large enthalpy of fusion

• Latent heat energy storage systems

• Decouple energy supply and demand → increase efficiency

• Wide range of applications from -40 to 500◦C, i.e. space to photovoltaics

PCM

DonatoRubinetti

Introduction

Application

Modeling

PhysicalModel

NumericalModel

ApplicationExample

Final Remarks

3

IntroductionThe need for modeling PCM

• Obtain fundamental understanding for freezing and melting cycle

• Predict the complex behavior well enough

• Efficiently choose among the vast selection of suitable PCM

• Design improvements

• Reduce development costs

PCM

DonatoRubinetti

Introduction

PhysicalModel

Couplings

NumericalModel

ApplicationExample

Final Remarks

4

Physical ModelMultiphysical couplings

Heat Transfer Fluid Flow

Geometry

Phase Change

Natural Convection

PCM

DonatoRubinetti

Introduction

PhysicalModel

NumericalModel

Implementation

Melted fraction

ApplicationExample

Final Remarks

5

Numerical ModelImplementation into COMSOL Multiphysics

Laminar Flow (CFD)

Heat Transfer in Fluids

auxiliary algebraicequations for material

properties

PCM

DonatoRubinetti

Introduction

PhysicalModel

NumericalModel

Implementation

Melted fraction

ApplicationExample

Final Remarks

6

Numerical ModelMelted fraction θ(T )

θ(T ) =

0, solidT−(Tm−∆T/2)

∆T, mushy

1, liquid

(1)

Temperature T [°C]M

elt F

ract

ion

θ(T

)

0.0

0.5

1.0

35.4 36.4 37.4

Tm

Tm + 1

2 ∆T

Tm − 1

2 ∆T

PCM

DonatoRubinetti

Introduction

PhysicalModel

NumericalModel

ApplicationExample

Temperature

Crossection 1

Results

Crosssection 2

Thermostat

Wall coretemperature

Final Remarks

7

Application ExampleObserved week in Oslo, temperature profile

Operation / Day Time [h]

Tem

pera

ture

T [°

C]

0 24 48 72 96 120 144 168

01/0

4/18

02/0

4/18

03/0

4/18

04/0

4/18

05/0

4/18

06/0

4/18

07/0

4/18−10

−5

05

10

Weather Forecast Oslo. 2018. url: https://www.wunderground.com/weather/no/oslo

https://www.wunderground.com/weather/no/oslo

PCM

DonatoRubinetti

Introduction

PhysicalModel

NumericalModel

ApplicationExample

Temperature

Crossection 1

Results

Crosssection 2

Thermostat


Final Remarks

8

Application ExampleWall crossection of a typical Norwegian building

9 15 15 50 50 120 100

x [mm] 21

1 2 3 4 5 6 7

1

2

Thermostat: Keep 19 < T < 26 °C+ internal heat gains

Weather data for T

PCM

DonatoRubinetti

Introduction

PhysicalModel

NumericalModel

ApplicationExample

Temperature

Crossection 1

Results

Crosssection 2

Thermostat


Final Remarks

9

Application ExampleResults - energy savings?

PCM

DonatoRubinetti

Introduction

PhysicalModel

NumericalModel

ApplicationExample

Temperature

Crossection 1

Results

Crosssection 2

Thermostat


Final Remarks

10

Application ExampleA less instulated wall crosssection

9 50 50 100

x [mm] 2

1

Indo

orro

om(a

ir)

PCM

CLT

woo

d

Am

bien

t(a

ir)

100 C

LT w

ood

1

PCM

DonatoRubinetti

Introduction

PhysicalModel

NumericalModel

ApplicationExample

Temperature

Crossection 1

Results

Crosssection 2

Thermostat


Final Remarks

11

Application ExampleResults - substantial energy savings!

Operation Time [h]

Roo

m H

eatin

g

0 24 48 72 96 120 144 168

Off

On

w/o PCM (66%)w/ PCM (13%)

PCM

DonatoRubinetti

Introduction

PhysicalModel

NumericalModel

ApplicationExample

Temperature

Crossection 1

Results

Crosssection 2

Thermostat


Final Remarks

12

Application ExampleWall core temperature


Wal

l Tem

pera

ture

[°C

]

0 24 48 72 96 120 144 168

1518

.520

2325

Time−averaged Room Temperature without PCM

Time−averaged Room Temperature with PCM

Melting Temperature (Liquefaction / Heat Absorption)

Freezing Temperature (Liquefaction / Heat Release)

With PCMWithout PCMPhase State

0.0

0.2

0.4

0.6

0.8

1.0

Mel

t / L

iqui

d F

ract

ion

θ(t)

PCM

DonatoRubinetti

Introduction

PhysicalModel

NumericalModel

ApplicationExample

Final Remarks

13

Final Remarks

Discussion• Release heat to reduce heating demand• For well insulated walls → marginal savings!• But: PCM reduces peak temperatures on both extremes• Cold climates → main benefit in summer

Conclusion• Comprehensive and suitable modeling approach for phase change phenomena

developed• Rapid orientation whether a PCM meets thermal, technical and economic

requirements• Model shows the importance of including indoor dynamics to assess the PCM

potential• Numerically stable model, extendable to enhanced PCM

PCM

DonatoRubinetti

Introduction

PhysicalModel

NumericalModel

ApplicationExample

Final Remarks

14

Thank you for your attention!

PCM

DonatoRubinetti

Appendix

A - ModelingFunctions

Melt Fraction

Gaussian

Heat Capacity

Density

Th. Conductivity

Carman-Kozeny

Viscosity

Requirements

Mushy zone

2D Test-case

BC

Carman-Kozeny

Validation

C - Results

D -ApplicationExample

E - 2DTest-Case

F -DimensionlessEstimation

15

A - Modeling FunctionsMelt Fraction θ(T )

θ(T ) =

0, solidT−(Tm−∆T/2)

∆T, mushy

1, liquid

(2)


elt F

ract

ion

θ(T

)

0.0

0.5

1.0

35.4 36.4 37.4

Tm

Tm + 1

2 ∆T

Tm − 1

2 ∆T

PCM

DonatoRubinetti

Appendix


Melt Fraction

Gaussian

Heat Capacity

Density

Th. Conductivity

Carman-Kozeny

Viscosity

Requirements

Mushy zone

2D Test-case

BC

Carman-Kozeny

Validation

C - Results


E - 2DTest-Case


16

A - Modeling FunctionsGaussian Distribution Function D(T )

D(T ) =e−

(T − Tm)2

(∆T/4)2√π(∆T/4)2

(3)

Temperature T [°C]G

auss

ian

Dis

trib

utio

n D

(T)

0.0

0.5

1.0

35.4 36.4 37.4

Tm

34.9 37.9

∆T = 2 K

∆T = 3 K

PCM

DonatoRubinetti

Appendix


Melt Fraction

Gaussian

Heat Capacity

Density

Th. Conductivity

Carman-Kozeny

Viscosity

Requirements

Mushy zone

2D Test-case

BC

Carman-Kozeny

Validation

C - Results


E - 2DTest-Case


17

A - Modeling FunctionsModified Heat Capacity Cp(T )

Cp(T ) = Cp,s+θ(T )(Cp,l−Cp,s)+D(T )L(4)

Temperature T [°C]H

eat C

apac

ity C

P(T

) [J

/(kg

K)]

1926

2400

35.4 36.4 37.4

Cp,l

Cp,s

∆T

D(T) L

PCM

DonatoRubinetti

Appendix


Melt Fraction

Gaussian

Heat Capacity

Density

Th. Conductivity

Carman-Kozeny

Viscosity

Requirements

Mushy zone

2D Test-case

BC

Carman-Kozeny

Validation

C - Results


E - 2DTest-Case


18

A - Modeling FunctionsModified Material Density ρ(T )

ρ(T ) = ρs+θ(T )(ρl−ρs)(5)

780

820

860

900


odifi

ed D

ensi

ty ρ

(T)

[kg/

(m^3

K)]

35.4 36.4 37.4

ρs

ρl

Tm − 1

2 ∆T

Tm + 1

2 ∆T

Tm

PCM

DonatoRubinetti

Appendix


Melt Fraction

Gaussian

Heat Capacity

Density

Th. Conductivity

Carman-Kozeny

Viscosity

Requirements

Mushy zone

2D Test-case

BC

Carman-Kozeny

Validation

C - Results


E - 2DTest-Case


19

A - Modeling FunctionsModified Thermal Conductivity k(T )

k(T ) = ks+k(T )(kl−ks)(6)

0.15

0.20

0.25

0.30

0.35

0.40


odifi

ed T

herm

al C

ondu

ctiv

ity k

(T)

[W/(

mK

)]

35.4 36.4 37.4

ks

kl

Tm − 1

2 ∆T

Tm + 1

2 ∆T

Tm

PCM

DonatoRubinetti

Appendix


Melt Fraction

Gaussian

Heat Capacity

Density

Th. Conductivity

Carman-Kozeny

Viscosity

Requirements

Mushy zone

2D Test-case

BC

Carman-Kozeny

Validation

C - Results


E - 2DTest-Case


20

A - Modeling FunctionsCarman-Kozeny Porosity Function S(T )

S(T ) = Am(1− θ(T ))2

θ(T )3 + ε(7)

0e+

004e

+08

8e+

08

Temperature T [°C]

Car

man

−K

ozen

y F

unct

ion

S(T

)

Solid dominatedmushy−zone

Liquid dominatedmushy−zone

35.4 36.4 37.4

Tm

Ali C. Kheirabadi and Dominic Groulx. “Simulating Phase Change Heat Transfer using COMSOL and FLUENT: Effect of the Mushy-ZoneConstant”. In: Computational Thermal Sciences: An International Journal 7.5-6 (2015). doi: 10.1615/ComputThermalScien.2016014279

https://doi.org/10.1615/ComputThermalScien.2016014279

PCM

DonatoRubinetti

Appendix


Melt Fraction

Gaussian

Heat Capacity

Density

Th. Conductivity

Carman-Kozeny

Viscosity

Requirements

Mushy zone

2D Test-case

BC

Carman-Kozeny

Validation

C - Results


E - 2DTest-Case


21

A - Modeling FunctionsViscosity of n-eicosane µ(T )

µ(T ) = (9×10−4 T 2−0.6529 T+119.94)×10−3

(8)

40 50 60 70 80 90

0.00

00.

001

0.00

20.

003

0.00

4

Temperature T [°C]D

ynam

ic V

isco

sity

µ(T

) [P

a s]

Tm

n − eicosanewater

PCM

DonatoRubinetti

Appendix


Melt Fraction

Gaussian

Heat Capacity

Density

Th. Conductivity

Carman-Kozeny

Viscosity

Requirements

Mushy zone

2D Test-case

BC

Carman-Kozeny

Validation

C - Results


E - 2DTest-Case


22

A - Modeling FunctionsBasic numerical requirements to govern the physics of PCM

conservation equation solid fraction liquid fraction

continuity Xmomentum Xenergy X X

→ direct approach: two subdomains for liquid and solid fraction with front trackingalgorithm

PCM

DonatoRubinetti

Appendix


Melt Fraction

Gaussian

Heat Capacity

Density

Th. Conductivity

Carman-Kozeny

Viscosity

Requirements

Mushy zone

2D Test-case

BC

Carman-Kozeny

Validation

C - Results


E - 2DTest-Case


23

A - Modeling FunctionsAlternative approach: introduction of a mushy zone

• Idea: material properties are smeared out over an user-defined meltingtemperature range

• Method: use of porosity formulation, liquid and solid co-exist in the mushyzone• Benefits:

• avoid numerical singularities• use one single mesh• easy to implement

• Setback: highly mesh-dependent solution in terms of physical accuracy

PCM

DonatoRubinetti

Appendix


Melt Fraction

Gaussian

Heat Capacity

Density

Th. Conductivity

Carman-Kozeny

Viscosity

Requirements

Mushy zone

2D Test-case

BC

Carman-Kozeny

Validation

C - Results


E - 2DTest-Case


24

A - Modeling Functions2D Test-case

Ω 1 x 1 cm2

x

y

n-eicosane

𝑇𝑇1 = 55 °C

PCM

DonatoRubinetti

Appendix


Melt Fraction

Gaussian

Heat Capacity

Density

Th. Conductivity

Carman-Kozeny

Viscosity

Requirements

Mushy zone

2D Test-case

BC

Carman-Kozeny

Validation

C - Results


E - 2DTest-Case


25

A - Modeling FunctionsBoundary conditions and setup

x

y

❶

❷

❸

❹Ω

Boundary CFD Heat Transfer

❶

No slipwall

𝑻𝑻𝑹𝑹 = 𝟒𝟒𝟒𝟒,𝟓𝟓𝟓𝟓,𝟕𝟕𝟒𝟒 °𝐂𝐂

❷

Thermal insulation❸

❹

𝒈𝒈

Pressure constraint point, 𝒑𝒑 = 𝟒𝟒 𝐏𝐏𝐏𝐏

Transient study, 𝟒𝟒 < 𝒕𝒕 < 𝟏𝟏𝟒𝟒𝟒𝟒𝟒𝟒 𝐬𝐬

PCM

DonatoRubinetti

Appendix


Melt Fraction

Gaussian

Heat Capacity

Density

Th. Conductivity

Carman-Kozeny

Viscosity

Requirements

Mushy zone

2D Test-case

BC

Carman-Kozeny

Validation

C - Results


E - 2DTest-Case


26

A - Modeling FunctionsCarman-Kozeny porosity function S(T )

S(T ) = Am(1− θ(T ))2

θ(T )3 + ε(9)

0e+

004e

+08

8e+

08

Temperature T [°C]

Car

man

−K

ozen

y F

unct

ion

S(T

)

Solid dominatedmushy−zone

Liquid dominatedmushy−zone

35.4 36.4 37.4

Tm

Ali C. Kheirabadi and Dominic Groulx. “Simulating Phase Change Heat Transfer using COMSOL and FLUENT: Effect of the Mushy-ZoneConstant”. In: Computational Thermal Sciences: An International Journal 7.5-6 (2015). doi: 10.1615/ComputThermalScien.2016014279

https://doi.org/10.1615/ComputThermalScien.2016014279

PCM

DonatoRubinetti

Appendix


Validation

Results

Model Setup

Material properties

C - Results


E - 2DTest-Case


27

B - Comparison to Experimental DataResults

−30 −20 −10 0 10 20 30

010

2030

4050

60

Radial Distance r [mm]

Axi

al D

ista

nce

z [m

m]

SimulationExperiment

t = 2164 s

−30 −20 −10 0 10 20 30

010

2030

4050

60

Radial Distance r [mm]A

xial

Dis

tanc

e z

[mm

]


t = 6964 s

−30 −20 −10 0 10 20 30

010

2030

4050

60


Axi

al D

ista

nce

z [m

m] Simulation

Experiment

t = 11524 s

PCM

DonatoRubinetti

Appendix


Validation

Results

Model Setup

Material properties

C - Results


E - 2DTest-Case


28

B - Comparison to Experimental DataModel Setup

z

r

𝑻𝑻𝑹𝑹 = 45 °C

n-eicosane

𝑯𝑯 = 59.8 mm

𝟐𝟐𝑹𝑹 = 63.8 mm

𝑻𝑻𝑩𝑩 = 32 °C

Ω

PCM

DonatoRubinetti

Appendix


Validation

Results

Model Setup

Material properties

C - Results


E - 2DTest-Case


29

B - Comparison to Experimental DataMaterial properties of n-eicosane, comparison with water

n-eicosane watersolid liquid solid liquid

density ρ [kg m−3] 910 769 916 997thermal conductivity k [W m−1 K−1] 0.423 0.146 1.6 0.6heat capacity Cp [kJ kg−1 K−1] 1.9 2.4 2.1 4.2melting temperature Tm [◦C] 36.4 - 0 -latent heat of fusion L [kJ kg−1] 248 - 334 -

PCM

DonatoRubinetti

Appendix


Validation

C - Results

Grashof Number

Melt Fraction

Mesh sensisitvity

Melt fraction

Gr

Validation I

Validation II


E - 2DTest-Case


30

C - ResultsGrashof Number

PCM

DonatoRubinetti

Appendix


Validation

C - Results

Grashof Number

Melt Fraction

Mesh sensisitvity

Melt fraction

Gr

Validation I

Validation II


E - 2DTest-Case


31

C - ResultsMelt Fraction

PCM

DonatoRubinetti

Appendix


Validation

C - Results

Grashof Number

Melt Fraction

Mesh sensisitvity

Melt fraction

Gr

Validation I

Validation II


E - 2DTest-Case


32

C - ResultsMesh sensitivity - melting front prediction

0.000 0.002 0.004 0.006 0.008 0.010

0.00

00.

002

0.00

40.

006

0.00

80.

010

x−Coordinate [m]

y−C

oord

inat

e [m

]

100 s

300 s

500 s

700 s

900 s

Mesh cells25 x 2550 x 50100 x 100150 x 150

PCM

DonatoRubinetti

Appendix


Validation

C - Results

Grashof Number

Melt Fraction

Mesh sensisitvity

Melt fraction

Gr

Validation I

Validation II


E - 2DTest-Case


33

C - ResultsMelt fraction - curvature of melting front

PCM

DonatoRubinetti

Appendix


Validation

C - Results

Grashof Number

Melt Fraction

Mesh sensisitvity

Melt fraction

Gr

Validation I

Validation II


E - 2DTest-Case


34

C - ResultsLocal Grashof number - influence of natural convection

PCM

DonatoRubinetti

Appendix


Validation

C - Results

Grashof Number

Melt Fraction

Mesh sensisitvity

Melt fraction

Gr

Validation I

Validation II


E - 2DTest-Case


35

C - ResultsValidation case - setup

z

r

𝑻𝑻𝑹𝑹 = 45 °C

n-eicosane

𝑯𝑯 = 59.8 mm

𝟐𝟐𝑹𝑹 = 63.8 mm

𝑻𝑻𝑩𝑩 = 32 °C

Ω

Benjamin J. Jones et al. “Experimental and numerical study of melting in a cylinder”. In: International Journal of Heat and Mass Transfer49.15-16 (2006), pp. 2724–2738

PCM

DonatoRubinetti

Appendix


Validation

C - Results

Grashof Number

Melt Fraction

Mesh sensisitvity

Melt fraction

Gr

Validation I

Validation II


E - 2DTest-Case


36

C - ResultsValidation case - comparison to experimental data

−30 −20 −10 0 10 20 30

010

2030

4050

60


Axi

al D

ista

nce

z [m

m]


t = 2164 s

−30 −20 −10 0 10 20 30

010

2030

4050

60Radial Distance r [mm]

Axi

al D

ista

nce

z [m

m]


t = 6964 s

−30 −20 −10 0 10 20 30

010

2030

4050

60


Axi

al D

ista

nce

z [m

m] Simulation

Experiment

t = 11524 s

PCM

DonatoRubinetti

Appendix


Validation

C - Results


Internal Heat Gains,Hourly

Indoor Temperaturew/o PCM

Indoor Temperaturew/ PCM

Thermostat

Wall CoreTemperature

Wall Cross-Section

E - 2DTest-Case


37

D - Application ExampleInternal Heat Gains, Hourly

050

010

0015

00

Hour of day [h]

Sen

sibl

e H

eat G

ain

[W]

0:00 03:00 06:00 09:00 12:00 15:00 18:00 21:00 24:00

PeopleLightingEl. equipmentRadiationSum

Komite-SN/K-034. Bygningers energiytelse, Beregning av energibehov of energiforsyning (engl.: Energy performance of buildings, calculationof energy needs and energy supply). URL:https://www.standard.no/no/Nettbutikk/produktkatalogen/Produktpresentasjon/?ProductID=859500. 2016

https://www.standard.no/no/Nettbutikk/produktkatalogen/Produktpresentasjon/?ProductID=859500

PCM

DonatoRubinetti

Appendix


Validation

C - Results





Thermostat


Wall Cross-Section

E - 2DTest-Case


38

D - Application ExampleIndoor Temperature w/o PCM


Tem

pera

ture

T [°

C]

0 24 48 72 96 120 144 168

01/0

4/18

02/0

4/18

03/0

4/18

04/0

4/18

05/0

4/18

06/0

4/18

07/0

4/18

−10

−5

05

1015

2025 Comfort

TemperatureRange

AmbientIndoor



PCM

DonatoRubinetti

Appendix


Validation

C - Results





Thermostat


Wall Cross-Section

E - 2DTest-Case


39

D - Application ExampleIndoor Temperature w/ PCM


Tem

pera

ture

T [°

C]

0 24 48 72 96 120 144 168

01/0

4/18

02/0

4/18

03/0

4/18

04/0

4/18

05/0

4/18

06/0

4/18

07/0

4/18

−10

−5

05

1015

2025 Comfort

TemperatureRange

AmbientIndoor



PCM

DonatoRubinetti

Appendix


Validation

C - Results





Thermostat


Wall Cross-Section

E - 2DTest-Case


40

D - Application ExampleThermostat

Operation Time [h]

Roo

m H

eatin

g

0 24 48 72 96 120 144 168

Off

On

w/o PCM (66%)w/ PCM (13%)

PCM

DonatoRubinetti

Appendix


Validation

C - Results





Thermostat


Wall Cross-Section

E - 2DTest-Case


41

D - Application ExampleWall Core Temperature


Wal

l Tem

pera

ture

[°C

]

0 24 48 72 96 120 144 168

1518

.520

2325

Time−averaged Room Temperature without PCM

Time−averaged Room Temperature with PCM

Melting Temperature (Liquefaction / Heat Absorption)

Freezing Temperature (Liquefaction / Heat Release)

With PCMWithout PCMPhase State

0.0

0.2

0.4

0.6

0.8

1.0

Mel

t / L

iqui

d F

ract

ion

θ(t)

PCM

DonatoRubinetti

Appendix


Validation

C - Results





Thermostat


Wall Cross-Section

E - 2DTest-Case


42

D - Application ExampleWall Cross-Section

9 15 15 50 50 120 100

x [mm] 21

Indo

orro

om(a

ir)

Soft

woo

d

PCM

CLT

woo

d

Soft

woo

d

Min

eral

woo

l

Am

bien

t(a

ir)

1 2 3 4 5 6 7

PCM

DonatoRubinetti

Appendix


Validation

C - Results


E - 2DTest-Case

2D Test-Case

Couplings

2D Test-case


43

E - 2D Test-Case2D Test-Case

Ω 0.01 x 0.01 m2

𝑻𝑻𝑹𝑹

x

y

n-eicosane

PCM

DonatoRubinetti

Appendix


Validation

C - Results


E - 2DTest-Case

2D Test-Case

Couplings

2D Test-case


44

E - 2D Test-CaseMultiphysical couplings

Solid

𝑻𝑻𝑹𝑹 ≫ 𝑻𝑻𝒎𝒎

𝑻𝑻𝑹𝑹 > 𝑻𝑻𝒎𝒎

Liquid

𝑻𝑻𝑹𝑹: Boundary Temperature

𝑻𝑻𝒎𝒎: Melting Temperature

PCM

DonatoRubinetti

Appendix


Validation

C - Results


E - 2DTest-Case

2D Test-Case

Couplings

2D Test-case


45

E - 2D Test-Case2D Test-case

Ω 0.01 x 0.01 m2

𝑻𝑻𝑹𝑹= 40, 55, 70 °C

x

y

n-eicosane

PCM

DonatoRubinetti

Appendix


Validation

C - Results


E - 2DTest-Case


Mushy ZoneInvestigation

Scaling Variables

DimensionlessEquations

DimensionlessNumbers

Values For LiquidFraction

46

F - Dimensionless EstimationMushy Zone Investigation

Temperature T [°C]

Mel

t Fra

ctio

n θ(

T)

0.0

0.5

1.0

Tm−1/2∆T Tm Tm+1/2∆T

Solid dominated

mushy−zone

Liquid dominated

mushy−zone

0

1

2

3

4

5

6

7

8

9

10

PCM

DonatoRubinetti

Appendix


Validation

C - Results


E - 2DTest-Case



Scaling Variables




47

F - Dimensionless EstimationScaling Variables

x =x

Hy =

y

Hp =

p − prefρu0

2

t =u0t

Huuu =

uuu

u0T =

T − Tref

TR − Tref

Φv =

(H

u0

)2

Φv ∇ = H∇ D

Dt=

(H

u0

)D

Dt

PCM

DonatoRubinetti

Appendix


Validation

C - Results


E - 2DTest-Case



Scaling Variables




48

F - Dimensionless EstimationDimensionless Equations

∇ · uuu = 0 (10)

Duuu

Dt= −∇p +

[µ

u0ρH

]∇2uuu −

[gβ(TR − Tm)H

u02

](ggg

g

)(T − Tm) (11)

DT

Dt=

[k

u0ρHCp

]∇2T +

[µu0

ρHCp(TR − Tm)

]Φv (12)

PCM

DonatoRubinetti

Appendix


Validation

C - Results


E - 2DTest-Case



Scaling Variables




49

F - Dimensionless EstimationDimensionless Numbers

Brinkman Br =µu0

2

k∆Theat production visc. dissipation vs heat transport by cond.

Grashof Gr =gβ∆TH3

ν2buoyant forces vs viscous forces

Prandtl Pr =ν

αmomentum diffusivity vs thermal diffusivity

Rayleigh Ra =gβ∆TH3

αν= GrPr heat transport conv. vs cond.

Reynolds Re =ρu0H

µinertial vs viscous forces

PCM

DonatoRubinetti

Appendix


Validation

C - Results


E - 2DTest-Case



Scaling Variables




50

F - Dimensionless EstimationValues For Liquid Fraction

Dimensionless group TR

40 ◦C 55 ◦C 70 ◦C[µ

u0ρH

]=

1

Re1 1 1

[gβ(TR − Tm)H

u02

]=

Gr

Re2=

Ra

PrRe2266 1376 2486

[k

u0ρHCp

]=

1

RePr0.008 0.008 0.008

[µu0

ρHCp(TR − Tm)

]=

Br

RePr1.25 × 10−10 2.42 × 10−11 1.34 × 10−11

phase change materials - comsol multiphysics...pcm donato rubinetti appendix a - modeling functions...

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