Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions...
Transcript of Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions...
![Page 1: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/1.jpg)
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
![Page 2: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/2.jpg)
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
![Page 3: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/3.jpg)
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
![Page 4: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/4.jpg)
PCM
DonatoRubinetti
Introduction
PhysicalModel
Couplings
NumericalModel
ApplicationExample
Final Remarks
4
Physical ModelMultiphysical couplings
Heat Transfer Fluid Flow
Geometry
Phase Change
Natural Convection
![Page 5: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/5.jpg)
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
![Page 6: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/6.jpg)
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
![Page 7: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/7.jpg)
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
![Page 8: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/8.jpg)
PCM
DonatoRubinetti
Introduction
PhysicalModel
NumericalModel
ApplicationExample
Temperature
Crossection 1
Results
Crosssection 2
Thermostat
Wall coretemperature
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
![Page 9: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/9.jpg)
PCM
DonatoRubinetti
Introduction
PhysicalModel
NumericalModel
ApplicationExample
Temperature
Crossection 1
Results
Crosssection 2
Thermostat
Wall coretemperature
Final Remarks
9
Application ExampleResults - energy savings?
![Page 10: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/10.jpg)
PCM
DonatoRubinetti
Introduction
PhysicalModel
NumericalModel
ApplicationExample
Temperature
Crossection 1
Results
Crosssection 2
Thermostat
Wall coretemperature
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
![Page 11: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/11.jpg)
PCM
DonatoRubinetti
Introduction
PhysicalModel
NumericalModel
ApplicationExample
Temperature
Crossection 1
Results
Crosssection 2
Thermostat
Wall coretemperature
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%)
![Page 12: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/12.jpg)
PCM
DonatoRubinetti
Introduction
PhysicalModel
NumericalModel
ApplicationExample
Temperature
Crossection 1
Results
Crosssection 2
Thermostat
Wall coretemperature
Final Remarks
12
Application ExampleWall core temperature
Operation / Day Time [h]
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)
![Page 13: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/13.jpg)
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
![Page 14: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/14.jpg)
PCM
DonatoRubinetti
Introduction
PhysicalModel
NumericalModel
ApplicationExample
Final Remarks
14
Thank you for your attention!
![Page 15: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/15.jpg)
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)
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
![Page 16: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/16.jpg)
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
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
![Page 17: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/17.jpg)
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
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
![Page 18: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/18.jpg)
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
18
A - Modeling FunctionsModified Material Density ρ(T )
ρ(T ) = ρs+θ(T )(ρl−ρs)(5)
780
820
860
900
Temperature T [°C]M
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
![Page 19: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/19.jpg)
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
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
Temperature T [°C]M
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
![Page 20: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/20.jpg)
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
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
![Page 21: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/21.jpg)
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
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
![Page 22: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/22.jpg)
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
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
![Page 23: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/23.jpg)
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
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
![Page 24: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/24.jpg)
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
24
A - Modeling Functions2D Test-case
Ω 1 x 1 cm2
x
y
n-eicosane
𝑇𝑇1 = 55 °C
![Page 25: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/25.jpg)
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
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
![Page 26: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/26.jpg)
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
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
![Page 27: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/27.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
Results
Model Setup
Material properties
C - Results
D -ApplicationExample
E - 2DTest-Case
F -DimensionlessEstimation
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
]
SimulationExperiment
t = 6964 s
−30 −20 −10 0 10 20 30
010
2030
4050
60
Radial Distance r [mm]
Axi
al D
ista
nce
z [m
m] Simulation
Experiment
t = 11524 s
![Page 28: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/28.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
Results
Model Setup
Material properties
C - Results
D -ApplicationExample
E - 2DTest-Case
F -DimensionlessEstimation
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
![Page 29: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/29.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
Results
Model Setup
Material properties
C - Results
D -ApplicationExample
E - 2DTest-Case
F -DimensionlessEstimation
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 -
![Page 30: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/30.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
Grashof Number
Melt Fraction
Mesh sensisitvity
Melt fraction
Gr
Validation I
Validation II
D -ApplicationExample
E - 2DTest-Case
F -DimensionlessEstimation
30
C - ResultsGrashof Number
![Page 31: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/31.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
Grashof Number
Melt Fraction
Mesh sensisitvity
Melt fraction
Gr
Validation I
Validation II
D -ApplicationExample
E - 2DTest-Case
F -DimensionlessEstimation
31
C - ResultsMelt Fraction
![Page 32: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/32.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
Grashof Number
Melt Fraction
Mesh sensisitvity
Melt fraction
Gr
Validation I
Validation II
D -ApplicationExample
E - 2DTest-Case
F -DimensionlessEstimation
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
![Page 33: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/33.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
Grashof Number
Melt Fraction
Mesh sensisitvity
Melt fraction
Gr
Validation I
Validation II
D -ApplicationExample
E - 2DTest-Case
F -DimensionlessEstimation
33
C - ResultsMelt fraction - curvature of melting front
![Page 34: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/34.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
Grashof Number
Melt Fraction
Mesh sensisitvity
Melt fraction
Gr
Validation I
Validation II
D -ApplicationExample
E - 2DTest-Case
F -DimensionlessEstimation
34
C - ResultsLocal Grashof number - influence of natural convection
![Page 35: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/35.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
Grashof Number
Melt Fraction
Mesh sensisitvity
Melt fraction
Gr
Validation I
Validation II
D -ApplicationExample
E - 2DTest-Case
F -DimensionlessEstimation
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
![Page 36: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/36.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
Grashof Number
Melt Fraction
Mesh sensisitvity
Melt fraction
Gr
Validation I
Validation II
D -ApplicationExample
E - 2DTest-Case
F -DimensionlessEstimation
36
C - ResultsValidation case - comparison to experimental data
−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
60Radial Distance r [mm]
Axi
al D
ista
nce
z [m
m]
SimulationExperiment
t = 6964 s
−30 −20 −10 0 10 20 30
010
2030
4050
60
Radial Distance r [mm]
Axi
al D
ista
nce
z [m
m] Simulation
Experiment
t = 11524 s
![Page 37: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/37.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
D -ApplicationExample
Internal Heat Gains,Hourly
Indoor Temperaturew/o PCM
Indoor Temperaturew/ PCM
Thermostat
Wall CoreTemperature
Wall Cross-Section
E - 2DTest-Case
F -DimensionlessEstimation
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
![Page 38: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/38.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
D -ApplicationExample
Internal Heat Gains,Hourly
Indoor Temperaturew/o PCM
Indoor Temperaturew/ PCM
Thermostat
Wall CoreTemperature
Wall Cross-Section
E - 2DTest-Case
F -DimensionlessEstimation
38
D - Application ExampleIndoor Temperature w/o PCM
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
1015
2025 Comfort
TemperatureRange
AmbientIndoor
Weather Forecast Oslo. 2018. url: https://www.wunderground.com/weather/no/oslo
![Page 39: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/39.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
D -ApplicationExample
Internal Heat Gains,Hourly
Indoor Temperaturew/o PCM
Indoor Temperaturew/ PCM
Thermostat
Wall CoreTemperature
Wall Cross-Section
E - 2DTest-Case
F -DimensionlessEstimation
39
D - Application ExampleIndoor Temperature w/ PCM
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
1015
2025 Comfort
TemperatureRange
AmbientIndoor
Weather Forecast Oslo. 2018. url: https://www.wunderground.com/weather/no/oslo
![Page 40: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/40.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
D -ApplicationExample
Internal Heat Gains,Hourly
Indoor Temperaturew/o PCM
Indoor Temperaturew/ PCM
Thermostat
Wall CoreTemperature
Wall Cross-Section
E - 2DTest-Case
F -DimensionlessEstimation
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%)
![Page 41: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/41.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
D -ApplicationExample
Internal Heat Gains,Hourly
Indoor Temperaturew/o PCM
Indoor Temperaturew/ PCM
Thermostat
Wall CoreTemperature
Wall Cross-Section
E - 2DTest-Case
F -DimensionlessEstimation
41
D - Application ExampleWall Core Temperature
Operation / Day Time [h]
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)
![Page 42: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/42.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
D -ApplicationExample
Internal Heat Gains,Hourly
Indoor Temperaturew/o PCM
Indoor Temperaturew/ PCM
Thermostat
Wall CoreTemperature
Wall Cross-Section
E - 2DTest-Case
F -DimensionlessEstimation
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
![Page 43: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/43.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
D -ApplicationExample
E - 2DTest-Case
2D Test-Case
Couplings
2D Test-case
F -DimensionlessEstimation
43
E - 2D Test-Case2D Test-Case
Ω 0.01 x 0.01 m2
𝑻𝑻𝑹𝑹
x
y
n-eicosane
![Page 44: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/44.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
D -ApplicationExample
E - 2DTest-Case
2D Test-Case
Couplings
2D Test-case
F -DimensionlessEstimation
44
E - 2D Test-CaseMultiphysical couplings
Solid
𝑻𝑻𝑹𝑹 ≫ 𝑻𝑻𝒎𝒎
𝑻𝑻𝑹𝑹 > 𝑻𝑻𝒎𝒎
Liquid
𝑻𝑻𝑹𝑹: Boundary Temperature
𝑻𝑻𝒎𝒎: Melting Temperature
![Page 45: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/45.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
D -ApplicationExample
E - 2DTest-Case
2D Test-Case
Couplings
2D Test-case
F -DimensionlessEstimation
45
E - 2D Test-Case2D Test-case
Ω 0.01 x 0.01 m2
𝑻𝑻𝑹𝑹= 40, 55, 70 °C
x
y
n-eicosane
![Page 46: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/46.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
D -ApplicationExample
E - 2DTest-Case
F -DimensionlessEstimation
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
![Page 47: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/47.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
D -ApplicationExample
E - 2DTest-Case
F -DimensionlessEstimation
Mushy ZoneInvestigation
Scaling Variables
DimensionlessEquations
DimensionlessNumbers
Values For LiquidFraction
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
![Page 48: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/48.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
D -ApplicationExample
E - 2DTest-Case
F -DimensionlessEstimation
Mushy ZoneInvestigation
Scaling Variables
DimensionlessEquations
DimensionlessNumbers
Values For LiquidFraction
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)
![Page 49: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/49.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
D -ApplicationExample
E - 2DTest-Case
F -DimensionlessEstimation
Mushy ZoneInvestigation
Scaling Variables
DimensionlessEquations
DimensionlessNumbers
Values For LiquidFraction
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
![Page 50: Phase Change Materials - COMSOL Multiphysics...PCM Donato Rubinetti Appendix A - Modeling Functions Melt Fraction Gaussian Heat Capacity Density Th. Conductivity Carman-Kozeny Viscosity](https://reader030.fdocuments.in/reader030/viewer/2022040722/5e3125768131a73ead3f695f/html5/thumbnails/50.jpg)
PCM
DonatoRubinetti
Appendix
A - ModelingFunctions
Validation
C - Results
D -ApplicationExample
E - 2DTest-Case
F -DimensionlessEstimation
Mushy ZoneInvestigation
Scaling Variables
DimensionlessEquations
DimensionlessNumbers
Values For LiquidFraction
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