Study of latent accumulator storage by heat transfer fluid (HTF) and external heat source (EHS)

Study of latent accumulator storage by heat transfer fluid (HTF) and external heat source (EHS)

Article in Physical and Chemical News · November 2012

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Phys. Chem. News 66 (2012) 56-68 PCN

STUDY OF LATENT ACCUMULATOR STORAGE

BY HEAT TRANSFER FLUID (HTF) AND EXTERNAL HEAT SOURCE (EHS)

B. Abouelkhayrat

¹

*, J. Lahjomri

¹

, A. Oubarra

¹

1 Hassan II University, Faculty of Sciences Ain Chock, Laboratory of Mechanics, 8 Km road to El Jadida PO Box 5366 Maarif, Casablanca, Morocco

* Corresponding author: E-mail: [email protected] Received: 20 June 2012; revised version accepted: 29 September 2012

Abstract:

Numerical study of thermal energy storage system is presented. The system under analysis consists of phase change material (PCM) encapsulated in a rectangular shape. In order to improve the latent heat accumulator as a thermal energy storage system, the heat transfer fluid (HTF) is associated with a heat external source (EHS) located at basis surfaces of the accumulator. By using enthalpy formulation, a two- dimensional model of the system is presented and validated by comparing numerical predictions with the analytical solution available in the literature. The paper discusses the effect of external source on time of complete melting of the PCM and the optimization of the source position. Results show that the selected location of the source is important to enhance heat transfer to PCM. The model is also used to assess the effects of various values of external heat flux, and the location area size of source on the system performance.

Keywords: Thermal energy storage system; Phase change material; External Heat Source; Latent heat; Accumulator.

1. Introduction

Recently, the thermal energy storage techniques are receiving increasing attention due to the imminent energy shortage and the high cost of energy. Therefore, it is necessary to store excess energy that would otherwise be wasted and also to bridge the gap between energy generation and consumption.

The storage of thermal energy as latent heat of fusion is advantageous over sensible heat because of its high storage density and the isothermal characteristics of charging and discharging process. Moreover, latent heat storage has the capacity to store heat of fusion at a constant or near temperature which correspond to the phase change material (PCM). In fact, there are large numbers of PCMs [1, 2] that melt and solidify at a wide range of temperatures, making them attractive in a number of applications [2].

The study of heat transfer characteristics of melting and solidification process is also one of the most attractive areas in the contemporary heat transfer research. In this problem often referred as a Stefan problem [3, 4], the analytical solutions are very limited due to the non-linearity nature at moving interface. Hence great attention has been given to the numerical solutions.

A survey of previously published papers dealing with latent heat thermal energy storage (LHTES) often reveals two geometries employed as PCM containers: rectangular [5-9] or cylindrical [10-13], for both opportunities of fluid flow, laminar [5, 6, 8, 10-12], and turbulent [7, 9],

with presence [7] or not [5, 6, 8, 9-12] of the natural convection when the temperature of PCM increases above the freezing temperature.

On account the unacceptably low thermal conductivity of most PCMs, the charging and discharging rates leading to slow. As a result, the enhancement techniques of heat transfer are required for most LHTES applications. Several studies have been conducted to study heat transfer enhancement techniques in PCMs and include finned tubes of different configurations [10, 14- 21] or using PCM dispersed with high conductivity particles [22], which leads to decrease the melting time and increasing the thermal conductivity.

Therefore, majority of the heat enhancement techniques have been based on the application of fins or particles embedded in the PCM. This is probably due to the simplicity, ease in fabrication and low cost of construction. The general observations drawn from these various studies demonstrate that there is decrease of densities of storage energy due to increase of numbers of fins or particles, added in the PCMs. However, in spite of improvement of melting time and the thermal conductivity, majority of classical accumulators does not satisfy the transfer kinetics for storage systems using the phase change materials discussed earlier.

In terms of performance of heat transfer

enhancement technique this paper proposes to

improve the transfer kinetics in LHTES without

affecting the mass density of PCM, by the study of a cell heated simultaneously by a classical heat transfer fluid and a heat source located on external wall of the accumulator. This available and free source can be solar energy or other industrial energy, wasted intermittently. Therefore, it is important to study, in addition, the melting process under the second kind boundary conditions. Liu and Ma [23] solved by numerical simulation the melting problem of a finite slab on which is imposed constant flux heating at its surface, and investigate the influences of various parameters on the melting process. This accumulator is subject only to effect of an external source without presence of the heat transfer fluid.

The main purpose of the present study is to optimize the position of the external heat source (EHS) in a classical accumulator by numerical simulation of melting time of PCM for various values of the dimensionless external flux applied on the absorbing area of different sizes.

2. Storage system

Basic geometry of latent heat thermal storage is shown schematically in Fig. 1. The storage system

is composed of two rectangular containers of PCM of thickness e

m

= ( R

3

− R

2

) separated by channel of thickness e

_f

= 2 R

₁

for the flowing heat transfer fluid (HTF). Due to the symmetry of the system about the x-axis only half plane is considered.

The HTF flows inside this channel and exchanges heat with the PCM through the wall of thickness e

_w

= ( R

₂

− R

₁

) . Moreover, heat transfer by HTF, the PCM is heated by a constant heat flux located on the upper surface. The system is thermally insulated except on the interval of width ∆ L = L

₂

− L

₁

where, a heat flux Φ

₀

is applied.

For the melting process, the PCM is initially solid, and its temperature is assumed at a certain value T

_i

below the melting point T

_mt

, while the HTF temperature T

₀

at the inlet of the storage unit was greater than melting temperature T

_mt

. These two conditions can be expressed mathematically as follows: ( T

_i

< T

_mt

) and ( T

₀

> T

_mt

) .

Figure 1: Physical model.

In order to simplify the problem such as the mathematical model can be easily formulated, we assume that:

∗ The flow is unidirectional and hydro- dynamically fully developed and corresponds to laminar Hagen-Poiseuille flow, with the viscous dissipation is negligible.

∗ PCM is assumed to be homogeneous and isotropic.

∗ All physical properties of both liquid and solid are constant.

∗ The volume expansion is neglected during the melting of the PCM. Densities and specific heat capacities are equal for both the solid and liquid phases by taking the average of the values for these two quantities between the solid and liquid phases.

∗ The effects of natural convection in the liquid

phase of the PCM are negligible. This hypothesis

is admitted for thin PCM thickness [7, 24]. In this

study, the aspect ratio is: ( R

₃

− R

₂

) / L = 0 . 13

Using the following dimensionless variables:

PecR

1

x = X ; R

1

y = Y ;

₂

1 l

R t

= a

τ ;

i 0

i

T T

T T

−

= −

θ (1)

( ) ( )

²

max

y U 1

y

u = U = −

;

1 s

* 0

R k .

∆ T

= Φ

Φ

(2)

The dimensionless equations of energy in different parts of system are written as:

For the fluid:

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

∂ θ + ∂

∂ θ

= ∂

∂ θ + ∂

τ

∂ θ

∂

2 2 f 2 2 f l 2

f f l f f

y x

Pec 1 a a x a

u a (3)

For the wall container:

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

∂ θ + ∂

∂ θ

= ∂ τ

∂ θ

∂

2 2 w 2 2 w l 2

w w

y x

Pec 1 a

a (4)

For the material (PCM):

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡

∂ θ + ∂

∂ θ

= ∂ τ

∂ + ∂ τ

∂ θ

∂

2 2 m 2 2 m l 2

m m

y x

Pec 1 a a f Ste

1 (5)

With

( 1 f ) a f a

a

_m

=

_s

− +

_l

(6) f

k ) f 1 ( k

k

_m

=

_s

− +

_l

(7)

f 1 max

a R

Pec = U ; [ ]

λ

= C

^m

T

⁰

− T

ⁱ

Ste (8) Such as: a

_f

, a

_w

, a

_s

and a

_l

are respectively the thermal diffusivities of the fluid, wall, and the material in solid and liquid state.

The liquid fraction f is estimated as:

) 11 ( if

1 f

) 10 ( if

1 f 0

) 9 ( if

0 f

mt m

mt m mt

mt m

ε + θ

≥ θ

=

ε + θ

≤ θ

≤ ε

− θ

≤

≤

ε

− θ

≤ θ

=

where:

θ mt : Dimensionless temperature of melting ε : Interval of the dimensionless melting The initial and boundary conditions are giving by the following expressions:

0 0 θ _m = θ _w = θ _f =

=

τ (12)

1 x 0

0 x

x

^{m w}

= θ

_f

=

∂ θ

= ∂

∂ θ

= ∂ (13)

x 0 x x Pec R

x L

^{m w f}

1

∂ = θ

= ∂

∂ θ

= ∂

∂ θ

= ∂ (14)

y 0 0

y

^f

=

∂ θ

= ∂ (15)

w f f w

f

y

k w k y

1

y θ = θ

∂ θ

= ∂

∂ θ

= ∂ (16)

m m w

w m 1 w

2

k y k y

R

y R θ = θ

∂ θ

= ∂

∂ θ

= ∂ (17)

[ [ ] ]

[ ]

⎢⎢

⎢⎢

⎢

⎣

⎡

Φ

∂ = θ

∈ ∂

∂ = θ

∪ ∂

∈

=

* 2 m

1 2 m 1

1 2

l y

; l x

y 0 l

; l l

; 0 x R

y R

(18)

The above equations represent: insulation of the accumulator, continuity of temperature and heat (fluid – wall; PCM – wall) and the external heat source.

Constant fluid temperature is specified at the inlet of the channel. At the exit, the fluid temperature gradient of the axial direction is set to zero. This particular boundary condition for the HTF is generally accepted in such studies. It reflects the fact that the fluid has already transferred its energy and reaches output section so that its temperature is invariant with respect to the axis of the flow. The computational domain at the exit is extended to ensure that the boundary conditions at the exit have no effect on the solutions.

3. Numerical solution and validation 3.1. Numerical procedure

In order to study the melting process, the above energy equations for fluid, wall container and PCM, were integrated by the method of alternating direction implicit (A.D.I). This is an implicit scheme with two steps of type prediction- correction. This method has a truncation error of order two in time and two in spaces,

( ^{∆ τ}

²

^{, ∆ x}

²

^{, ∆ y}

²

)

Ο and is unconditionally stable [25]. The solution of equations 3, 4 and 5, is based on the discretization equations that are solved iteratively by using the tri-diagonal matrix in two directions.

On the x direction, three matrixes are obtained, corresponding respectively to fluid, wall container and PCM domain. They are conducted to the temperature profile on each domain, and coupled their results with boundary conditions: fluid-wall container (see Eq. 16) and PCM-wall container (see Eq. 17). On the other hand, for the y direction, only hybrid matrix is obtained. It combines simultaneously the three domains as well as the boundary conditions surrounding the wall container.

By running several numerical codes with different number of the size of the time step, ∆ τ , and with various values of the space steps,

( ∆ x , ∆ y ) , we found that ∆ τ = 10

⁻⁵

, ∆ x = 10

⁻³

,

and ∆ y = 1 . 67 × 10

⁻³

are optimum. These values

give a good compromise between the time of computation and the variation of the solution at any point and ensure grid independence.

At each time step, the program is tested so that total energy supplied to accumulator from HTF Q

ⁱ_f^→o

and external source Q

_sc

, are equal to energy stored in the fluid Q

^st_f

, in the wall container Q

^st_w

and in the material PCM Q

^st_m

. The overall energy balance is expressed as:

stm stw stf o sc

if

Q Q Q Q

Q

^→

+ = + + (19) The terms which they appear in equation (19) are given for a unit width, as:

( ) d u [ 1 ( ,l y , ) ] dy T

Pec a R

Q k

¹ _f

0 12

l o f

if^→

= ∆ τ ∫ − θ τ

(

2 1

)

12

* l

sc s

. . R . Pec . T . d . x x a

Q = k Φ ∆ τ −

( ) ( )

[ x , y , d x , y , ] dxdy

T Pec a R

Q k

1 0

l

0 f f

12 f f stf

∫ ∫ ^{θ τ + τ − θ τ}

×

∆

=

( ) ( )

[ x , y , d x , y , ] dxdy T

Pec a R

Q k

1 2R R 1

l

0

w w

12 w st w w

∫ ∫ ^{θ τ + τ − θ τ}

×

∆

=

( ) ( )

[ ]

{

( ) ( )

[ f x , y , d f x , y , ] dxdy Ste

1

, y , x d , y , x T

Pec a R Q k

1 R3

1 2

R

R R l 0

m 2 m

m 1 st m m

⎭ ⎬ τ ⎫

− τ + τ +

τ θ

− τ + τ θ

∆

= ∫ ∫

3.2. Validation of numerical solution

Before presenting the numerical results for the phase change system, the numerical code must be validated by comparison with analytical solution available in the literature.

First, the phase-change model was checked against the classical Newman problem. A liquid PCM is initially at T

_i

> T

_mt

in an infinitely long rectangular with a uniform cross section. At t > 0 , the surface is kept at a temperature T

_e

< T

_mt

and solidification takes place. The dimensionless position of the solid-liquid interface versus dimensionless time is calculated by numerical code and compared to its dimensionless analytical values given by the function [3]:

( ) τ = 2 γ τ

y where γ is an unknown to be

determined by solving the following equation (20):

γρλ

⎟⎟ =

⎠

⎜⎜ ⎞

⎝

⎛ γ −

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ γ

− π

⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛ γ −

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ γ

− π

l 2

l i mt l

l

s 2

s e mt s

s

exp a erfc a

T T a k

exp a erf a

T T a k

(20)

Result of comparison between numerical and analytical solutions is reported in Fig. 2. It can be seen that the present model agrees well with the results of this study.

Figure 2: Dimensionless interface position of front with dimensionless time.

The second validation is for the case of rectangular container filled by PCM in solid state with a constant heat flux imposed on one surface, when the opposite surface is adiabatic.

Analytical dimensionless time τ of premelting for different values of dimensionless heat flux Φ

^*

is given by solving equation 21 realized by Zhongliang Liu and Chongfang Ma [23].

[ ( ) ]

¹

1 n

2 2

2

*

exp n

n 1 2 3

1

^{∞ −}

=

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ − π τ

− π + τ

=

Φ ∑ (21)

The numerical solution is tested also, against a variation of dimensionless melting starting time with dimensionless heat flux that is given by the analytical solution (Eq. 21). The results depicted in Fig. 3 show a very good agreement and prove that their reliability and accuracy are assured;

furthermore this numerical model can give the results to a wide range of heat flux.

0.0001 0.001 0.01 0.1 1 10 100 1000

1E-7 1E-6 0.00001 0.0001 0.001 0.01 0.1 1 10 100 1000 10000

Numerical solution Analytical solution [23]

Figure 3: Variation of dimensionless melting starting time with dimensionless heat flux.

4. The problem data

The study is conducted, using a rectangular accumulator filled of PCM having the dimensionless geometric parameters as follows:

5 . 1 R

R

_{3 1}

= , R

₂

R

₁

= 1 . 1 in height and

( PecR ) 3 L

l =

₁

= in length. The channel has dimensionless thickness unity, for the flowing of heat transfer fluid (HTF). This latter is separated of the PCM by the wall of dimensionless thickness: ( R

₂

− R

₁

) R

₁

= 0 . 1

Figure (1) is a schematic shape of the model storage heating where the left and right vertical walls are insulated. The top wall is adiabatic, except on a field width ∆ l = l n such as

( n = 1 , 2 , 3 , 4 , 5 , 10 , 20 ... ) where a heat flux Φ

₀

is applied (see Fig. (1)). This external heat source is

placed at various locations on top surface boundary of the accumulator for selecting the optimum position of the source.

The thermophysical properties of PCM, wall container and HTF are selected from available product [1]. The characteristics used to evaluate the thermal energy storage system studied are described in the table 1.

The data from this table provides the following dimensionless ratios, used in the numerical analysis.

79 . 0 a

a

_{f l}

= ; a

_s

a

_l

= k

_s

k

_l

= 2 . 06 ; 01

. 0 k

k

_{f w}

= ; a

_w

a

_l

= 81 . 57 ; k

_w

k

_l

= 101 . 89

Also, the numerical calculations of the problem

are performed for fixed values of dimensionless

melting temperature, Stefan and Peclet numbers,

respectively: θ

_mt

= 0 . 29 ; Ste = 0 . 5 ; Pec = 10 .

5. Results and discussion

In order to optimize the position of the external heat source (EHS), the minimum dimensionless melting time is adopted as a criterion, for various values of the dimensionless external flux applied on the location of size ∆ l = l n .

The dimensionless melting time, τ

_m

, is the time required to store the maximum dimensionless latent energy ( Q

_L

= q

_L

q

_LMax

= 1 ) in the

accumulator with or without EHS, where q

_LMax

is the total latent heat stored in the accumulator.

The location of EHS is limited by two axial coordinates: x

₁

and x

₂

, such as:

n l l x

x

₂

−

₁

= ∆ = (22) While the middle of this location area is identified as:

n 2 l x n 2 l x

M

₀

=

₁

+ =

₂

− (23)

PCM Wall

container

Fluid

Compound CaCl2.6H2O Steel H

2

O

Melting temp (

^o

C) 29.9 --- ---

Heat of fusion λ (kJ/kg) 187 --- --- C

l

2.2

Specific heat capacity

C(kJ. kg

^-1

K

^-1

) C

s

1.4 0.465 4.18

K

l

0.53 Thermal conductivity

K(Wm

^-1

K

^-1

) K

s

1.09 54 0.6

ρ

l

1530

Density ρ(kg.m

^-3

) ρ

s

1710 7833 1000

Table 1: Dimensional parameters employed in the numerical analysis.

The achievement of the numerical study is executed with the same conditions for both accumulators: with and without external heat source (EHS); then it is possible to compare their numerical results.

Of how well future outcomes are likely to be predicted for the same conditions of model used, the results obtained provide some correlations in the context of Excel as statistical model; with coefficient of determination is equal to 0.99.

5.1. Optimum position of the external heat source First, the minimum dimensionless melting time required to store the maximum of latent energy in accumulator without EHS deducted from simulation is: τ

_mt

= 0 . 78 .

For accumulator with EHS, an identical dimensionless heat flux ( ^Φ

^*

^{= 10} ) is applied for same step equals l/300 of source shift on constant location size ∆l=l/20. The dimensionless melting time is calculated for any source shift.

Figure 4 shows the variation of dimensionless melting time, with shift of middle of external heat source. The examination of this figure reveals that each displacement of EHS has different effect on dimensionless time τ

_mt

required to store the maximum latent energy. As the thermal conductivity of PCM is low, the purpose of source creates another field melted coupled with that produced by the fluid flow. The study is initiated

by the source positions situated close of input section. Its effect associated with the fluid regarding the reduction of the melting of the material is limited; because a large part of PCM is not affected to the heat flux from EHS and the HTF. In contrast, the positions of the source away from input section involves an obvious improvement of the melting time. It is explained by the advent of higher thermal levels in the intermediate parts of areas melted. Therefore the opposition of these two zones has an effect to improve heating in middle portion between them.

Gradually, as position of the source moves away from the fluid inlet, the heating of the material which still solid by melted areas becomes more important. Eventually the time required for melting the PCM improves. Therefore, the effect of the source was advantageous when its position is increasingly away from input section of the accumulator. Then it will result a decrease in the melting time. Middle of heat external source M

0

=2.45 is displacement selected as optimum, which is near the exit section of the fluid flow.

This optimum position of EHS corresponds to the

minimum dimensionless melting time τ

_mt

= 0 . 5 .

Then, to store the same amount of energy in

accumulator with EHS, the minimum time has an

improvement of about 36% compared to one

without heat source.

D im ens ionl ess m e lt ing ti m e

Locations of External Heat Source

Figure 4: Variation of dimensionless melting time with locations of external heat source.

5.2. Effect of various values of heat external flux The previous study is repeated for the same size of source ∆ l = l 20 such as various values of dimensionless heat external flux is applied from

*

= 5

Φ to100. The first observation reached, is that the optimum position is always opposite to inlet heat flow for almost the same position of EHS:

05 . 0 45 . 2

M

₀

= ±

Figure 5 shows the influence of various values of dimensionless flux of source on dimensionless

melting time τ

_mt

, where the increase in values of 

the dimensionless flux, Φ

^*

, involves a decrease in the minimum time τ

_mt

. So, it is clear that heat external source (EHS) has a significant influence on the melting time for storing the same amount of energy. From the data depicted in Fig. 5, equation 24 is concluding for predicting the dimensionless melting time τ

_mt,

for the values of the dimensionless flux, Φ

^*

, from 5 to 100.

( ) ^Φ

^∗

×

−

=

τ

_mt

0 . 869 0 . 16 ln (24) The table 2 depicted the results for computations of melting time gain G

_mt

. The calculation formula to achieve the gain is as follows:

Figure 5: Influence of various values of Φ

^*

on dimensionless melting time.

100

G

_withoutEHS

mt

withoutEHS withEHS mt

mt mt

×

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ τ

τ

−

= τ , (25)

when the values of Φ

^*

increases, the gain is increased simultaneously.

Φ∗ 5 10 20 30 40 50 60 70 80 90 100

18% 36% 48% 56% 61% 66% 70% 73% 75% 77% 79%

Table 2: Melting time gain.

5.3. Location size of the external heat source Occasionally the energy provided, is important only for a limited period, it is therefore necessary to exploit it at maximum. It is possible to study this similar problem, for accumulator with EHS, with an identical dimensionless heat flux ( Φ

^*

= 10 ) applied on various locations of size ∆ l = l n , for different values of n from 1 to 20.

Figure 6 shows the influence of location size on displacement of optimum position. It is seen from this below figure, that the location size has a significant influence on the displacement of middle of the optimum position. This latter moves from the right to the center of the accumulator,

gradually as the size becomes broader. This minimizes the maximum of the melting time, which explains the preference of an optimal position that provides a more expands heating. In front of a low thermal conductivity, the improvement of thermal transfer by extending the heated area accelerates the melting and subsequently minimizes the melting time.

The data obtained are used to conclude the following equation (26) to determine the middle of external heat source position (M

0

) for any location size:

( ) ^l _n ^{0 . 319} ( ) ^l _n ^{0 . 094} ( ) ^l _n ^{2 . 47 ( 26 )}

081 . 0

M

₀ = ³− ²− +

Figure 6: Influence of location size on optimum position.

From this study it is also possible to establish the variations of dimensionless melting time with location size. Figure 7 shows the effect of the source size associated with the action of a fluid on the melting time. From the previous study, the increasing in the source size acts favorably on the transfer kinetics and hence melting time reducing.

This is due to the material area increasingly heated especially as the source size is larger. It is clear from the curve shown in Figure 7 that the effect of increasing the source size is more remarkable for low flux values applying on small size. On the

other hand, the further increase in size for large sources is less important due to the slight variation of dimensionless melting time. In the limiting case of sources that apply on a large part of the lateral section, the widening of the source has no effect.

In the limit of the sources that have larger sizes,

the heat is transferred to the side sections insulated

on the left and right which explains the reason

why dimensionless melting time is almost

invariant.

The equation 27 provides the dimensionless melting time whatever the size of opening on which dimensionless heat flux is applied.

( ) ( ) ( )

( ) ^l _n ^{1 . 492} ( ) ^l _n ^{1 . 263} ( ) ^l _n ^{0 . 662}

022 . 1

l n 373 . n 0 065 l . n 0 004 l . 0

2 3

4 5

6 mt

+

− +

−

+

−

=

τ

(27)

Figure 7: Influence of location size on dimensionless melting time.

5.4. Isothermals of accumulator without and with heat external source in optimal position

For the same conditions cited in the previous subsection (5.1), the isotherms diagram depict in Fig. 8 the melting interface by dotted lines at different dimensionless times ( τ = 0.06, 0.12, 0.3 and 0.52) for two accumulators, one without EHS (in the left of Fig. 8) and the other with a localized source in the optimal position (at right of Fig. 8).

For both, the initial temperature of the PCM is lower than its melting temperature. In the accumulator without EHS, the melting process is started by heat transfer of HTF; while the accumulator on which is imposed a heating constant flux at its upper surface, the melting process effected simultaneously in two opposite area: by the external heat source where it is localized and in the entrance of fluid, as shown in following Fig. 8.

First, the presence of heat source affect the shape of isotherms especially for high times due to apparition of isotherms associated to the location of source. These isotherms are circulars and penetrate advantage in the accumulator when the time increases. Furthermore, comparing the second Fig. 8.b and the third Fig. 8.c shows that the melting area is larger in the accumulator with EHS. So, when the time varies, the melting area of the PCM has become more important in the accumulator with EHS than without source. This explains the decrease of minimum melting time, for the same energy stored, in accumulator with EHS. Thus store the same amount of energy with EHS is faster. Without source, energy is transmitted to the material by the fluid close to the inlet section and the remaining diffuses into the accumulator. Where the source is present in the

optimal position, it operates on the opposite zones of the input section that is means where the effect of the heating fluid is not transferred because of the low conductivity of the PCM. The kinetics of heat transfer is greater. The area of the PCM with high thermal level is doubled: the first is due to the effect of HTF, while the second is associated with the presence of the source. Therefore, melting time is smaller compared with the case of accumulator without source.

After the complete melting of the PCM in accumulator with EHS as shown in Fig. 8.d, the evolution of latent heat is stabilized, and the heat transferred from the source will be stored in the form of sensible heat. Temperature level is more important than the case without EHS so heat stored will be greater. The following sub paragraph discusses in detail the evolution of different forms of energies.

5.5. Latent, sensible and total energy

In order to estimate the effect of EHS for the optimal position, it is recommended to study in details, different forms of energy: latent (Fig. 9), sensible (Fig. 10) and total energy (Fig. 11).

Since the evolution of melting time with different values of heat flux is reduced rapidly for

*

≤ 20

Φ , (see Fig. 5) followed by a slow change, heat external flux ( Φ

^*

=0 and 10) are selected to compare their corresponding dimensionless energy

stored: sensible ⎟

⎠

⎜ ⎞

⎝

⎛ =

SMax S

q

S

q

Q , latent

⎟ ⎠

⎜ ⎞

⎝

⎛ =

LMax L

q

L

q

Q and total heat

( )

( ) ^⎟ _⎠ ^⎞

⎜ ⎝

⎛ = + +

LMax SMax

L

T

q

S

q q q

Q These

energies are calculated in reference to theirs maximums values without EHS, as:

( )

⎟⎠

⎜ ⎞

⎝

⎛ −

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

τ θ

=

∫ ∫

1 2 1 3 R R

R R l

0 m

S

R R R R

l

dy . dx . , y , x Q

1 3

1 2

( )

⎟⎠

⎜ ⎞

⎝

⎛ −

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

τ

=

∫ ∫

1 2 1 3 R R

R R l

0 L

R R R R

l

dy . dx . , y , x f Q

1 3

1 2

( ) ( )

[ ]

(

+

)

⎜⎝⎛ − ⎟⎠⎞

⎥⎥

⎥⎥

⎦

⎤

⎢⎢

⎢⎢

⎣

⎡

τ +

τ θ

= −

∫ ∫

−

1 2 1 1 3 R

R

R R l

0 m 1

T

R R R Ste R 1 l

dy . dx . , y , x f Ste , y , x Q

1 3

1 2

.

For figures on the left For figures on the right

Φ*=0, Ste=0.5 , Pec=10 and ∆l=l/20 Φ*=10 , Ste=0.5 , Pec=10 and ∆l=l/20

0.00 0.50 1.00 1.50 2.00 2.50 3.00

0.00 0.50 1.00 1.50

0.00 0.50 1.00 1.50 2.00 2.50 3.00

0.00 0.50 1.00 1.50

Fig. 8.a: Dimensionless time τ

₁

= 0.06

0.00 0.50 1.00 1.50 2.00 2.50 3.00

0.00 0.50 1.00 1.50

0.00 0.50 1.00 1.50 2.00 2.50 3.00

0.00 0.50 1.00 1.50

Fig. 8.b: Dimensionless time τ

₂

= 0.12

0.00 0.50 1.00 1.50 2.00 2.50 3.00

0.00 0.50 1.00 1.50

0.00 0.50 1.00 1.50 2.00 2.50 3.00

0.00 0.50 1.00 1.50

Fig. 8.c: Dimensionless time τ

₃

= 0.3

0.00 0.50 1.00 1.50 2.00 2.50 3.00

0.00 0.50 1.00 1.50

0.00 0.50 1.00 1.50 2.00 2.50 3.00

0.00 0.50 1.00 1.50

Fig. 8.d: Dimensionless time τ

₄

= 0.54 Figure 8: Isotherms diagram.

Moreover, the complete melting for accumulator with EHS is the condition in this study. Figure 9 shows the profiles of dimensionless latent energy, for two values of dimensionless heat flux Φ

^*

= 0 (accumulator without EHS) and Φ

^*

=10 (accumulator with EHS). As results, for Φ

^*

=10 and τ = 0 . 5 , the ratio

of latent energy stored has reached the maximum=0.99.

The energy curves moves upward

simultaneously in favor of the accumulator with

EHS versus the other case; however in the end, the

latent energy stored in accumulator with EHS

remains changed slightly. This means the PCM

has reached the melting. In absence of EHS, the

melting is not complete at the same duration. This is explained by the above Fig. (8.d) of isothermal at time great than 0.5, which depicts the presence of melting interface in accumulator without EHS.

Storing latent energy at same duration in accumulator with and without EHS has an improvement of about 20% compared to the one without source.

0.00 0.10 0.20 0.30 0.40 0.50

Dimensionless time 0.00

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Dimensionless Latent Energy

Dimensionless Latent Energy

with EHS ( =10) without EHS ( = 0)

φ∗

Ste=0.5 and Pec=10

Figure 9: Evolution of dimensionless latent heat with dimensionless time.

The sensible heat term cannot be neglected, when studying the latent heat accumulator as a thermal energy storage system especially, some periods of melting is marked by pure sensible heat.

Figure 10 shows the evolution of dimensionless sensible heat with dimensionless time for two case of accumulator: with heat flux Φ*= 0 (accumulator without EHS) and Φ*=10 (accumulator with EHS). This energy increases

according to three phases: Firstly, at the beginning of melting time, the solid area is predominant.

Secondly, when the major part of PCM, has a temperature of fusion, the sensible heat takes the values slightly unchanged. Third, at the completion of melting time, the liquid area is predominant. When the source is present the increase of sensible energy at τ=0.5 reached 30%

compared to the one without source.

0.00 0.10 0.20 0.30 0.40 0.50

Dimensionless time 0.00

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80

Dimensionless Sensible Energy

Dimensionless Sensible Energy

with EHS ( =10) without EHS ( = 0)

φ∗

Ste=0.5 and Pec=10

Figure 10: Evolution of dimensionless sensible heat with dimensionless time.

It is important to evaluate the overall types of the heat, but the important parameter used to evaluate a thermal energy storage system is the total energy Q

T

stored that is composed by sensible Q

S

and latent heat Q

L

.

Figure 11 shows the evolution of dimensionless total energy stored with dimensionless time. The total energy is increased simultaneously for both accumulators.

Examination of Fig. 11 reveals that the presence of heat external source at the accumulator produces essentially the improvement of heat storage for value of Φ*= 10, comparatively with accumulator without external source for a same duration. These curves show an improvement of this quantity that reached 28%, for accumulator with EHS in optimum position compared to the one without source.

0.00 0.10 0.20 0.30 0.40 0.50

Dimensionless time 0.00

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

Dimensionless Total Energy

Dimensionless Total Energy with EHS ( =10) without EHS ( = 0)

φ∗

Ste=0.5 and Pec=10

Figure 11: Evolution of total dimensionless energy stored with dimensionless time.

6. Conclusion

An accumulator using phase change material with configuration in Fig. 1 has been studied numerically. Phase change heat transfer of the PCM is solved simultaneously with the flow heat transfer and external heat source placed at upper surface of accumulator.

The performance of storage system has been evaluated by examining the effect of the source location on melting time of PCM. Results show that use of external free and available source improve the kinetics of transfer without affecting the density of the PCM.

References

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Nomenclature

R distance separating different parts Subscripts

e thickness (m)

i

initially

L length of accumulator(m)

0

Inlet/external dimensionless length

e

external dimensionless space coordinate

S

Sensible/solid dimensionless space coordinate

L

Latent

X space coordinate (m)

T

Total

Y space coordinate m

sc

source

T Temperature (K)

m

material PCM

dimensional time (s)

_ϖ

wall container

thermal diffusivity (m

²

/s)

^f

Fluid thermal conductivity (W/mK)

mt

melting

heat capacity (J.kg

^-1

K

^-1

)

Max

maximum Φ

0

heat flux (W/m

²

)

Φ

^*

dimensionless heat flux Superscripts

liquid fraction * refers to dimensionless quantities

dimensionless energy i Inlet

q Energy (J) o outlet

dimensionless velocity st Stored integer number

middle of the location area Abbreviations:

G Gain Pec Peclet number

a

f

R U

_{max 1}

=

∆T

T0−Ti

Greek symbols Ste Stefan number [ ]

λ

ⁱ

m

T T

C −

=

⁰

θ dimensionless temperature PCM Phase Change Material τ dimensionless time HTF Heat Transfer Fluid λ latent heat of fusion(kJ/kg)

ε Interval of the dimensionless melting

LHTES Latent Heat Thermal Energy Storage

γ unknown number EHS External Heat Source