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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

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


Latent heat thermal energy storage systems



The studied composite materials

KL1_250 KD1

These composites are introduced by V. Canseco, Y. Anguy, J. J. Roa and E. Palomo in[1].



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



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



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



Dimensions of the samples

6.84mm

9mm

6.84mm

10.6mm



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



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



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)



Initial and boundary conditions

T0 = 311.9C

TD = 322C



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



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



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)



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



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



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



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

1

Dim

ensi

onle

ss T

t=5e-2s

t=182s




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

1

Dim

ensi

onle

ss T

KL1_250 KD1homogene



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.



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



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



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



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



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



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.



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



initial initial

α = 0.9 α = 0.9



α = 0.8 α = 0.8

α = 0.7 α = 0.7



α = 0.6 α = 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 0.2 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



Conclusions

Enlarging the pores allow to

accelerate the melting process inside the ports.

increasing the heat flux while keeping a constant thermal conductivity.



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



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


Dilatations de KL1_250 dans la direction x

α = 0.9 α = 0.8 α = 0.7

α = 0.6



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


Dilatations de KD1 dans la direction x

α = 0.9 α = 0.8 α = 0.7

α = 0.6



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.



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

1

Dim

ensi

onle

ss T

homogene



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



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


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


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



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



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

1

Dim

ensi

onle

ss T

homogene



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



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


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


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



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.


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.

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