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Asymptotic properties of carbon based compositematerials used in latent heat thermal energy storage
systems
Maimouna Mint Brahim
Laboratoire TREFLE-I2M, Université de Bordeaux
LMAC, Université de technologie de Compiègne
16 Décember, 2015
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Sommaire
1. Introduction
2. The studied composite materials
3. The model
4. The time scheme
5. Initial and boundary conditions
6. The analytic solution in the case of a homogeneous PCM
7. Comparison with a homogeneous PCM
8. Effects of increasing the effective thermal conductivity
9. Influence of the pores size
10. Influence of the pores shape
11. Bibliographie
Maimouna Mint Brahim I2M - LMAC 2 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Latent heat thermal energy storage systems
Maimouna Mint Brahim I2M - LMAC 3 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
The studied composite materials
KL1_250 KD1
These composites are introduced by V. Canseco, Y. Anguy, J. J. Roa and E. Palomo in[1].
Maimouna Mint Brahim I2M - LMAC 4 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Thermo-physical properties of the composites
Parameter KL1_250 KD1porosity(%) 82 70κPCM (W /m/K) 1 1CpPCM (J /kg/K) 4544 4544CpFoam (J /kg/K) 1402 1402ρPCM (J /kg/K) 1634 1634ρFoam (J /kg/K) 1318 1318∆hm (KJ /kg) 480 480
the porosity of a composite is defined as E =VporesVTotal
We want κeff the same for both compositesWe know that κeff = a ∗ κFoam + b
Maimouna Mint Brahim I2M - LMAC 5 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
The effective thermal conductivity of the composites
200 400 600 800 1000
κ matrix
(W/m/K)
0
10
20
30
40
50
κeff
ecti
ve (
W/m
/K)
KL1_250
KD1
336 958
Maimouna Mint Brahim I2M - LMAC 6 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Thermal energy storage capacity
For 2 different 2D rectangular samples, of dimensions: x1 × y1 and x2 × y2 with porositiesE1 and E2 with the same latent heat of fusion ∆hm, their thermal energy storagecapacities are
Q1 = E1x1y1∆hmand
Q2 = E2x2y2∆hmhence to have Q1 = Q2 with x1 = x2 we need to have E1y1 = E2y2
In our case EKL1_250 = 82% > EKD1 = 70%we need to take yKD1 > yKL1_250
Maimouna Mint Brahim I2M - LMAC 7 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Dimensions of the samples
6.84mm
9mm
6.84mm
10.6mm
Maimouna Mint Brahim I2M - LMAC 8 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
The model
∂tH (T )− div(κ∇T ) = 0 in ΩG ∪ ΩS
κ∂nT = 0 on ΓN
T = TD on ΓD
[κ∂nT ] = 0 and γ
T (0, x) = T0 in Ω
The enthalpy H (T ) is given by:
H (T ) =
[(ρc)sS (1− f (T )) + (ρc)lS f (T )]T + ρS∆hmf (T ) in ΩS
(ρc)GT in ΩG
Maimouna Mint Brahim I2M - LMAC 9 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
The time scheme
Implicit Euler : H n+1−H n∆t − div(κ∇T n+1) = 0
Chernoff Scheme : H = β−1(T ) where T = β(H ) → ∂tH = 1/β′(T )∂tT
on définie:H n+1 = H n + γ(T n+1 − β(H n))
where γ corresponds to 1β′(T )
γ
∆t(T n+1 − β(H n))− div(κ∇T n+1) = 0
Maimouna Mint Brahim I2M - LMAC 10 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Linear ProblemλE(u)− div(κ∇u) = g dans ΩG ∪ ΩS
κ∂nu = 0 sur ΓN
u = uD sur ΓD
[κ∂nu] = 0 et R(κ∂nu)s = [u] sur γ
Sobolov spacesH (div,Ω) := q ∈ L2(Ω)2 : divq ∈ L2(Ω)
H0,N (div,Ω) := q ∈ H (div,Ω) : q.n = 0 sur ΓN
Mixed Formulationp = κ∇u
Find (u, p) such as∫Ω
1
κp.qdx +
∫Ωu div q dx −
∫γR(p.n)(q.n) dγ =
∫ΓD
uDq.ndγ ∀q ∈ H0,N (div,Ω)
∫ΩλE(u)vdx −
∫Ωv div p dx =
∫Ωvgdx ∀v ∈ L2(Ω)
Maimouna Mint Brahim I2M - LMAC 11 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Initial and boundary conditions
T0 = 311.9C
TD = 322C
Maimouna Mint Brahim I2M - LMAC 12 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
The analytic solution in the case of a homogeneous PCM
T (x, t) = TD + (Tm − TD)erf ( x
2√αt
)
erf (ξ)0 < x ≤ X (t)
with X (t) = 2ξ√αt where ξ is solution to
Stefan
erf (ξ)eξ2=√πξ
with
Stefan =c(TD − Tm)
∆hm
The heat flux at surface y = 0 is given by
q(0, t) = −κ(Tm − TD)
erf (ξ)√παt
The similarity variable is defined by µ = x√αt
Maimouna Mint Brahim I2M - LMAC 13 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
PCM with equivalent thermo-physical properties
Recall : the porosity of a composite is defied as E =VporesVTotal
If ρG and ρS are the density of the matrix and the PCM then ρequivalent = EρS + (1− E)ρG
CpG and CpS the thermal capacities of the matrix and of the PCM,
Cpequivalent = ECpS + (1− E)CpG
∆hm is the latent heat of fusion of the PCM contained in the matrix, ∆hmequivalent = E∆hm
Maimouna Mint Brahim I2M - LMAC 14 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Dimensionless Variables
x∗ =x
xmax, t∗ =
αt
x2, T ∗ =
T − TDTm − TD
, X ∗ =X
xmaxand q∗ =
xmaxq
κ(Tm − TD)
T ∗(x∗, t) =erf ( x∗
2√t∗
)
erf (ξ), X ∗(t) = 2ξ
√t∗, q∗(0, t) = −
1√πerf (ξ)
√t∗
For a composite material,we introduce the mean dimensionless temperature
T ∗Mean(y∗, t) =
1
Nx
n=Nx∑n=1
T ∗(x, y, t)
Maimouna Mint Brahim I2M - LMAC 15 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Comparison with a homogeneous PCM
KL1_250
0 0.2 0.4 0.6 0.8 1yd
0
2
4
6
8
10
Tgraphite-Tsalt(°C)
t=5e-2s
t=182s
KD1
0 0.2 0.4 0.6 0.8 1yd
0
2
4
6
8
10
Tgraphite-Tsalt(°C)
t=5e-2s
t=182s
Maimouna Mint Brahim I2M - LMAC 16 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
KL1_250: temperature profile
KL1_250
0 0.2 0.4 0.6 0.8 1yd
312
314
316
318
320
322
324
T(°C)
t=182s
t=5e-2s
KD1
0 0.2 0.4 0.6 0.8 1yd
312
314
316
318
320
322
324
T(°C)
t=180s
t=5e-2s
Maimouna Mint Brahim I2M - LMAC 17 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
KL1_250
0 0.5 1 1.5 2Similarity variable
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
t=5e-2s
t=182s
KD1
0 1 2 3 4 5Similarity variable
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
t=5e-2st=182s
Maimouna Mint Brahim I2M - LMAC 18 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
KL1_250
0 0.2 0.4 0.6 0.8 1ERF(Similarity variable)
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
t=182s
t=5e-2s
KD1
0 0.2 0.4 0.6 0.8 1ERF(Similarity variable)
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
t=5e-2s
t=182s
Maimouna Mint Brahim I2M - LMAC 19 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
0 0.5 1 1.5 2Similarity variable
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
KL1_250 KD1
0 0.2 0.4 0.6 0.8 1ERF(Similarity variable)
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
KL1_250 KD1homogene
Maimouna Mint Brahim I2M - LMAC 20 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
0 1 2 3 4 51/sqrt(time)
0
300
600
900
1200
1500
1800
Hea
t fl
ux a
t x=
0
KD1 KL1_250homogene
0 0.2 0.4 0.6 0.8sqrt(timeD)
0
0.002
0.004
0.006
Mel
ting F
ront
Posi
tion
KD1 KL1_250
homogene
Conclusion : from t = 182s KL1_250 could be assimilated to a homogeneous PCM withequivalent thermo-physical properties.
Maimouna Mint Brahim I2M - LMAC 21 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Effects of increasing the effective thermal conductivity
κeffective (W /m/K) κFoam KL1_250 κFoamKD110 410 12520 958 33630 1510 58040 2060 84850 2615 1128
Maimouna Mint Brahim I2M - LMAC 22 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
KL1_250
0 0.1 0.2 0.3 0.4 0.5Similarity variable
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
homogene
κeff
=50
κeff
=40
κeff
=30
κeff
=20
κeff
=10
KD1
0 0.5 1 1.5 2 2.5 3Similarity variable
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
homogene
κeff
=50
κeff
=40
κeff
=30
κeff
=20
κeff
=10
Maimouna Mint Brahim I2M - LMAC 23 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
KL1_250
0 0.1 0.2 0.3 0.4 0.5Erf(Similarity variable)
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
homogene
κeff
=50
κeff
=40
κeff
=30
κeff
=20
κeff
=10
KD1
0 0.2 0.4 0.6 0.8 1Erf(Similarity variable)
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
homogene
κeff
=50
κeff
=40
κeff
=30
κeff
=20
κeff
=10
Maimouna Mint Brahim I2M - LMAC 24 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
KL1_250
0 0.2 0.4 0.6 0.8 1sqrt(timeD)
0
0.002
0.004
0.006
0.008
Mel
ting F
ront
Posi
tion
homogene
κeff
=50
κeff
=40
κeff
=30
κeff
=20
κeff
=10
KD1
0 0.2 0.4 0.6 0.8sqrt(timeD)
0
0.002
0.004
0.006
Mel
ting F
ront
Posi
tion
homogene
κeff
=50
κeff
=40
κeff
=30
κeff
=20
κeff
=10
Maimouna Mint Brahim I2M - LMAC 25 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
KL1_250
0 0.2 0.4 0.6 0.8 11/sqrt(time)
0
200
400
600
800
1000
1200
1400
1600
1800
Hea
t fl
ux a
t x=
0
homogene
κeff
=50
κeff
=40
κeff
=30
κeff
=20
κeff
=10
KD1
0 0.2 0.4 0.6 0.8 11/sqrt(time)
0
200
400
600
800
1000
1200
1400
1600
1800
Hea
t fl
ux a
t x=
0
homogene
κeff
=50
κeff
=40
κeff
=30
κeff
=20
κeff
=10
Maimouna Mint Brahim I2M - LMAC 26 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
KL1_250
10 20 30 40 50κ
eff(W/m/K)
920
930
940
950
960
970
980
Char
gin
g t
ime(
s)
KD1
10 20 30 40 50κ
eff(W/m/K)
100
150
200
250
300
350
400
Char
gin
g t
ime(
s)
Conclusions :
Changing the values of the effective thermal conductivity has more effects in thecase of KD1 because the volume occupied by the matrix represents 30% of thetotal volume where for KL1_250 it represents only 18% of the total volume.
Increasing the value of the effective thermal conductivity allows to accelerate thecharging/discharging cycles.
Maimouna Mint Brahim I2M - LMAC 27 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Influence of the pores size
Dilatation in both directions :fnew(x, y) = f (αx, αy) with α ∈ 0.6, 0.7, 0.8, 0.9, 1
κeff = 20 W /m/K , ∀α
Maimouna Mint Brahim I2M - LMAC 28 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
initial initial
α = 0.9 α = 0.9
Maimouna Mint Brahim I2M - LMAC 29 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
α = 0.8 α = 0.8
α = 0.7 α = 0.7
Maimouna Mint Brahim I2M - LMAC 30 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
α = 0.6 α = 0.6
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
0 0.2 0.4 0.6 0.8 11/sqrt(time)
0
100
200
300
400
500
600
700
Hea
t fl
ux a
t x=
0
homogene
Initialalpha=0.9
alpha=0.8
alpha=0.7
alpha=0.6
0 0.2 0.4 0.6 0.8 11/sqrt(time)
0
100
200
300
400
500
600
700
Hea
t fl
ux a
t x=
0
homogene
Initialalpha=0.9
alpha=0.8
alpha=0.7
alpha=0.6
0 0.1 0.2 0.3 0.4sqrt(timeD)
0
0.001
0.002
0.003
0.004
0.005
0.006
Mel
ting F
ront
Posi
tion(m
)
homogene
Initialalpha=0.9
alpha=0.8
alpha=0.7
alpha=0.6
0 0.1 0.2 0.3 0.4sqrt(timeD)
0
0.001
0.002
0.003
0.004
0.005
0.006
Mel
ting F
ront
Posi
tion(m
)
homogene
Initialalpha=0.9
alpha=0.8
alpha=0.7
alpha=0.6
Maimouna Mint Brahim I2M - LMAC 32 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Conclusions
Enlarging the pores allow to
accelerate the melting process inside the ports.
increasing the heat flux while keeping a constant thermal conductivity.
Maimouna Mint Brahim I2M - LMAC 33 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Influence of the pores shape
Dilatations in both directions :fnouveau(x, y) = f (αx, βy) avec α, β ∈ 0.6, 0.7, 0.8, 0.9, 1
κeff = 20 W /m/K , ∀α, β
Maimouna Mint Brahim I2M - LMAC 34 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Dilatations de KL1_250 dans la direction y
β = 0.9 β = 0.8 β = 0.7 β = 0.6Maimouna Mint Brahim I2M - LMAC 35 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Dilatations de KL1_250 dans la direction x
α = 0.9 α = 0.8 α = 0.7
α = 0.6
Maimouna Mint Brahim I2M - LMAC 36 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Dilatations de KD1 dans la direction y
β = 0.9 β = 0.8 β = 0.7 β = 0.6Maimouna Mint Brahim I2M - LMAC 37 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Dilatations de KD1 dans la direction x
α = 0.9 α = 0.8 α = 0.7
α = 0.6
Maimouna Mint Brahim I2M - LMAC 38 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
KL1_250 KD1
0.6 0.7 0.8 0.9 1dilatation coefficient
200
250
300
350
400
450
500
κ(W
/m/K
)
dilatation in the x-directiondilatation in the y-direction
For KL1_250 we can see that the more we poll in the y direction the more weneed to increase the thermal conductivity of the matrix and the more we poll inthe x direction the more we increase the thermal conductivity of the matrix.
On note the opposite for KD1.
Explication : for KL1_250 les pores s’étalent dans la direction y et on sollicite danscette direction → on doit augmenter κmatrice.
Maimouna Mint Brahim I2M - LMAC 39 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Résultats pour KL1_250
0 0.2 0.4 0.6 0.8 1Similarity variable
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
homogene
Initialbeta=0.9beta=0.8beta=0.7beta=0.6
0 0.2 0.4 0.6 0.8 1Similarity variable
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
homogene
Initialalpha=0.9
alpha=0.8
alpha=0.7
alpha=0.6
0 0.2 0.4 0.6 0.8 1ERF(Similarity variable)
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
homogene
Initialbeta=0.9beta=0.8beta=0.7beta=0.6
0 0.2 0.4 0.6 0.8 1ERF(Similarity variable)
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
homogene
Initialalpha=0.9
alpha=0.8
alpha=0.7
alpha=0.6
Maimouna Mint Brahim I2M - LMAC 40 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Résultats pour KL1_250
0 0.2 0.4 0.6 0.8 11/sqrt(time)
0
100
200
300
400
500
600
700
Hea
t fl
ux a
t x=
0
homogene
Initialbeta=0.9beta=0.8beta=0.7beta=0.6
0 0.2 0.4 0.6 0.8 11/sqrt(time)
0
100
200
300
400
500
600
700
Hea
t fl
ux a
t x=
0
homogene
Initialalpha=0.9
alpha=0.8
alpha=0.7
alpha=0.6
0 0.1 0.2 0.3 0.4sqrt(timeD)
0
0.001
0.002
0.003
0.004
0.005
0.006
Mel
ting F
ront
Posi
tion
homogene
Initialbeta=0.9beta=0.8beta=0.7beta=0.6
0 0.1 0.2 0.3 0.4sqrt(timeD)
0
0.001
0.002
0.003
0.004
0.005
0.006
Mel
ting F
ront
Posi
tion
homogene
Initialalpha=0.9
alpha=0.8
alpha=0.7
alpha=0.6
Maimouna Mint Brahim I2M - LMAC 41 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Résultats pour KD1
0 0.5 1 1.5 2 2.5 3Similarity variable
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
homogene
Initialbeta=0.9beta=0.8beta=0.7beta=0.6
0 0.5 1 1.5 2Similarity variable
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
homogene
Initialalpha=0.9
alpha=0.8
alpha=0.7
alpha=0.6
0 0.2 0.4 0.6 0.8 1ERF(Similarity variable)
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
homogene
Initialbeta=0.9beta=0.8beta=0.7beta=0.6
0 0.2 0.4 0.6 0.8 1ERF(Similarity variable)
0
0.2
0.4
0.6
0.8
1
Dim
ensi
onle
ss T
homogene
Initialalpha=0.9
alpha=0.8
alpha=0.7
alpha=0.6
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Résultats pour KD1
0 0.2 0.4 0.6 0.8 11/sqrt(time)
0
100
200
300
400
500
600
700
Hea
t fl
ux a
t x=
0
homogene
Initialbeta=0.9beta=0.8beta=0.7beta=0.6
0 0.2 0.4 0.6 0.8 11/sqrt(time)
0
100
200
300
400
500
600
700
Hea
t fl
ux a
t x=
0
homogene
Initialalpha=0.9
alpha=0.8
alpha=0.7
alpha=0.6
0 0.1 0.2 0.3 0.4sqrt(timeD)
0
0.001
0.002
0.003
0.004
0.005
0.006
Mel
ting F
ront
Posi
tion
homogene
Initialbeta=0.9beta=0.8beta=0.7beta=0.6
0 0.1 0.2 0.3 0.4sqrt(timeD)
0
0.001
0.002
0.003
0.004
0.005
0.006
Mel
ting F
ront
Posi
tion
homogene
Initialalpha=0.9
alpha=0.8
alpha=0.7
alpha=0.6
Maimouna Mint Brahim I2M - LMAC 43 / 45
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Intro Foams Model Scheme SimCond SolutAnal CompHomogene CondTh Poresize PoreShape Biblio
Conclusions
En dilatant dans la direction x
Pour KL1_250 on remarque des variations dans les profiles de températures ainsiqu’aux flux surfaciques puisqu’on augmente la valeur de κmatrice.
Pour KD1 on a presque le même comportement pour les différents α puisqu’on nesollicite pas dans cette direction.
En dilatant dans la direction y on constate que dans les deux cas les fronts de fusionsvont plus vites puisque les pores se sont allongés dans le sens où les matériaux sont"chauffés".Donc pour accélérer les cycles de charges/décharges il faudrait dilater les pores dans lesens du gradient de température.
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Bibliographie
V. Canseco, Y. Anguy, J. J. Roa, and E. Palomo, Structural and mechanicalcharacterization of graphite foam/phase change material composites, Carbon, 74(2014), pp. 266–281.
R. M. Christensen, Mechanics of composite materials, Courier Corporation, 2012.
H. Jopek and T. Strek, Optimization of the effective thermal conductivity of acomposite, INTECH Open Access Publisher, 2011.
A. L. Kalamkarov, I. V. Andrianov, V. V. Danishevsâ, et al., Asymptotichomogenization of composite materials and structures, Applied Mechanics Reviews,62 (2009), p. 030802.
A. L. Kalamkarov and K. S. Challagulla, Effective properties of composite materials,reinforced structures and smart composites: Asymptotic homogenization approach,in Effective Properties of Heterogeneous Materials, Springer, 2013, pp. 283–363.
V. Morisson, Heat transfer modelling within graphite/salt composites: from thepore scale equations to the energy storage system, PhD thesis, Bordeaux 1, 2008.