Pressure effects in the planar solidification of a finite slab: convective cooling and prescribed...

IL NUOVO CIMENTO VOL. 19 C, N. 2 Marzo-Aprile 1996

Pressure effects in the planar solidification of a finite slab: convective cooling and prescribed heat flux boundary conditions

M. CONTI, F. MARINOZZI and L. PAPAGNO

Dipartimento di Matematica e Fisica, Universitd di Camerino - 62032 Camerino, Italy

(ricevuto il 28 Ottobre 1994; revisionato il 19 Giugno 1995; approvato il 13 Luglio 1995)

Summary. - - The classical Stefan problem assumes a f'Lxed melting temperature. However, when the solid phase is the one with lower density (e.g., water) the solidification of the system causes an overall volume increase that is often contrasted by the container walls. In that case the growing pressure determines a continuous lowering of the freezing point, and the temperature field as well as the interface motion are strongly affected. This paper is concerned with these aspects of the problem; the planar solidification of a slab of finite thickness, contrasted by an opposing elastic force, is numerically simulated. The effects of two different boundary conditions are analysed. When the solidification is driven by convective cooling, the continuous advancement of the melting front is replaced by an asymptotic behaviour, until thermal equih'brium is attained. When the boundary condition is specified in terms of a prescribed heat flow, the melting front velocity is slowed down by a growing adverse temperature gradient. The influence of various parameters on the process is presented and discussed.

PACS 91.35.Dc - Heat flow; geothermy. PACS 91.60.Hg - Phase changes. PACS 64.70.Dv - Solid-liquid transitions.

Nomenclature

a thermal diffusivity Bi Blot number, defined as: Bi = hLo ~ks c specific-heat capacity E Young modulus of the elastic opposing wall F non-dimensional parameter , defined as: F = ( 1 / ~ ) ( 1 / Q s - 1 / ~ I ) ( 1 - Q s / Q l ) "

�9 E / ( y E + 1) G non-dimensional parameter , defined as: G = Tmo/O for the convective-cooling

problem, or G = Tmo/O' for the prescribed-heat-flux problem, respectively h convective-heat-transfer coefficient k thermal conductivity L length of the PCM slab

257

258 M. CONTI, F. MARINOZZI and L. PAPAGNO

Lo initial length of the PCM slab p pressure P0 inital pressure of the PCM St Stefan number, defined as: St = csO/~ for the convective-cooling problem, or

St = cs 0'/)~ for the prescribed-heat-flux problem, respectively T temperature T a temperature of the cooling fluid To initial temperature of the PCM T~ actual melting temperature of the PCM T~ 0 initial melting temperature of the PCM T dimensionless ~mperature, defined as: T = ( T - T~o)/O for the convective-cool-

ing problem or T = (T - Tin0)/0' for the prescribed-heat-flux problem, respectively t time U internal energy per unit volume U dimensionless internal energy, defined as: U = (U-Qsc~Tmo)/(Ql~) V volume of the PCM Vo initial volume of the PCM x coordinate direction xf coordinate of the melting front

Greek symbols ~] effective compressibility of the PCM 0 temperature reference for the convective-cooling problem: 0 = Tm0 - Ta 0' temperature reference for the prescribed-heat-flux problem: O' = Lo v /ks )~ PCM latent heat

dimensionless coordinate direction, defined as: ~ = x/Lo ~f dimensionless coordinate of the melting front ~) density a liquid fraction of the PCM v dimensionless time, defined as: v = as t /L 2 X non-dimensional variable, defined as: • = k p / k s

heat flux extracted at the surface x = 0

Subscripts f melting front 1 liquid phase of the PCM p PCM s solid phase of the PCM 0 initial conditions

1 . - I n t r o d u c t i o n

Stefan problems describing a solid-liquid phase change attracted a great deal of interest for many decades. The physical domain in which the heat equation must be solved is a priori unknown. This results in non-linear interface conditions. Due to these difficulties, few exact closed-form solutions are available [1]. Over the years, a number of approximate analytical solutions have been produced, using per turbat ive expansions[2,3], variational methods[4], and the integral heat balance concept[5].

PRESSURE EFFECTS IN THE PLANAR SOLIDIFICATION ETC. 259

A large number of numerical techniques have been developed, based on the enthalpy weak formulation of the problem [6], the isotherm migration [7] and the appearent heat capacity methods [8]. Extensive reviews of the available literature before 1990 can be found in Yao and Prusa[9] and Lunardini[10].

A full description of the problem should incorporate the effects due to the different density of the liquid and solid phases. Analytical [11,12] and numerical [13, 14] contributions on this subject were mainly concerned with the kinematic coupling of the volume change and the heat transfer process. All these studies assume a fixed melting temperature.

However, the density change accompanying the phase transition affects the pressure field, both through the flow of the liquid phase that is induced inside the system, and through the interactions with the container walls when the Phase Change Material (PCM) expands during the phase transition. In these cases the pressure dependence of the freezing point must be taken into account. The first effect has been treated by Charach and Rubinstein [15] in an analytical paper, showing that the initial singularity of the interface velocity, typical for the solutions of the classical Stefan problem with Dirichlet boundary conditions, is regularized by the dynamic pressure field.

On the other side, pressure-temperature effects due to the interaction of the PCM with the domain walls are of central relevance in geological processes. Sliding of glaciers over their beds, for instance, is interpreted in terms of the ,,melting and regelation- process[16]. The problem is to explain how ice, assumed to be at the melting point, moves past bumps in the glacier bed. The excess pressure on the upstream sides of bumps allows melting of ice on these sides. The melt water will flow round the bump and refreeze on the downstream side, where the pressure is lower. To maintain the process, the latent heat of fusion released during solidification is conducted upstream through the bump and is used for melting. Moreover, an active area of research is concerned with the strong dynamical effects that occur in water-saturated soils due to the growth of the expanding solid phase: enormous forces can be developed, capable of displacing retaining walls, lifting pavements, fenceposts, and utility poles[17]. Similar problems are faced in other areas of engineering applications: when the phase transition is driven in a cyclic way, as for instance in thermal storage applications, the melting PCM can be enclosed by the solid phase, and hence hindered to expand [18].

Recently, the pressure-temperature effects due to the interaction of the PCM with the container walls have been addressed by Conti[19] in a numerical study. The problem was treated under prescribed-temperature (Dirichlet) boundary conditions. A planar geometry was considered; the pressure and temperature fields were coupled through the Clapeyron equation. This paper is an extension of the previous study, and takes into account different boundary conditions: the phase transition process is assumed to be driven by either a prescribed heat flux or a convective cooling, respectively. The system is shown in fig. 1: a finite slab, initially in the liquid phase, is assumed to expand upon freezing. The process is contrasted by a force that represents, according to the Hooke's law, the elastic behaviour of the container wall: the increasing pressure determines a continuous lowering of the freezing point. A convective term arises in the heat equation, due to the expansion of the PCM; this effect leads to a correction in the interface motion given by the product of the relative density change and the Stefan number [9]. Usually, both these parameters are rather small and the correction is quite small too. Hence the pure diffusion equation has been


utilized to treat the heat transfer inside the PCM. No pressure dependence of the PCM latent heat is assumed; equal values of the effective compressibility are assumed for the two phases.

The problem has been formulated through the weak-solution approach usually referred to as the enthalpy method; the resulting equation has been solved numerically by the finite-difference method.

The results that will be presented show that the interface motion is strongly affected by the pressure increase; when the solidification is driven by convective cooling, the continuous advancement of the melting front is replaced by an asymptotic behaviour, until thermal equilibrium is attained. When the boundary condition is specified in terms of a prescribed heat flow, the melting velocity is slowed down by a growing adverse temperature gradient.

2. - F o r m u l a t i o n o f the problem

The system is illustrated in fig. 1, where the emerging solid phase is represented by the shaded area and a spring simulates the elastic behaviour of the opposing container wall. Let L0, Vo be the initial length and volume of the PCM slab. For times t -< 0 the PCM is in the liquid phase at a uniform temperature T(x, 0) = To > Tin0, where Tmo is the melting temperature at the initial pressure Po. For times t ~> 0 two different problems will be considered:

i) convective-cooling problem: in this case the surface at x = 0 is suddenly exposed to a cooling fluid, at a constant temperature Ta < Tm0,

ii) prescribed-heat-flux problem: here the solidification process is driven by extracting a constant heat flux ~ at the surface x = 0.

Adiabatic wall at x --Lo is assumed. A solid layer grows adjacent to the surface x -- 0 and advances with time. The volume increase of the PCM, due to the lower density of the solid phase, is allowed in the x-direction only, i.e. perfectly rigid lateral walls are assumed. The opposing wall causes a pressure increase on the PCM given, in

Fig. 1. - Schematic diagram of the slab and the opposing elastic wall. The shaded area represents the emerging solid phase; the spring simulates the elastic behaviour of the opposing wall at x=L.


the elastic limit, by

(1) P - Po = E ( V - Vo) Vo '

where p, V indicate the actual pressure and volume of the PCM and E is the Young modulus of the elastic wall. This choice of the elastic constant is due to the one- dimensional character of the problem. The PCM volume is determined by solving (1) in conjunction with

(2) P - Po = 1 ( V - V ' )

V'

Equation (2) describes the compression of the PCM; r] is the effective compressibility of the two-phase mixture and V' is the volume that would be occupied by the PCM in the absence of opposing forces. Notice that (V' - Vo) is the net volume increase due to the phase change when the melting front is at x = xf; mass conservation implies (V' - V0) = xf - (xfQs)/r whereas V0 = L0. Uniform and unit cross-section of the system is assumed. Thus, it is easy to show that

(3) V'=Vo[I+(1-6)--~s) xf ] el ~oo "

When ( 1 - Qs/Q1)(xf/Lo) is small, as is usually the case, eqs. (2) and (3) give

(4) P - P 0 = ~ ]E+I Q z Lo

Equation (4) shows that the pressure of the system increases as the solidification goes on; as a consequence, the melting temperature changes according to the Clapeyron equation

(5) dp

dTm Tm (~)l "-1 - ~)s 1 )

If the PCM latent heat is not affected by the pressure variation, eq. (5) can be easily integrated to give

(6) Tm = Tmo'exp[ (~[1_r ( p - Po)]"

In the following it will be assumed that:

i) the heat transfer is only due to conduction, i.e. the kinematic effects of the phase change are neglected;

ii) the energy interaction due to mechanical work is neglected too.

It means that density variations during the process are ignored except insofar as they give rise to a pressure increase.


Under these assumptions, the one-dimensional heat equation can be written as

@U _ O kp %T (7) %t %x 3x

U in eq. (7) indicates the energy stored per unit volume, and is related to the temperature via

(8) T = a U + fi,

where

a=(osc~) -1 , f l=O, {U<<.OscsTm},

a = O , f i = T m , {O<<- U - o s e s T m <<-1} 01;~

a = (OlCi) -1 ~ = Tm(X - Oscs ) ~ { U - oscsTm >1} - - , �9

' 01cl cl 011

The initial and boundary conditions for eq. (7) are specified by

T(x, O) = To, __%T I = O. OX [ x = LO

The condition at x = 0 is given by

aT ~:o hEY(O, t) - Y:] = ks -~x (convective-cooling problem),

The heat flow ~ is defined as positive when extracted from the PCM. Introducing the temperature scales 0 = Tmo- Ta for the convective-cooling

problem and O' = Lo ~P/ks for the prescribed-heat-flux problem, the equations can be conveniently restated in terms of the following non-dimensional variables:

= x/Lo, v = ast/L 2 , l] = (V - OscsTmo)/(Ol~),

Os

and

= (T - Tmo)/O ,

= (T - - Tmo)/O' ,

1)(1 01 r / E + l t ' '

St = cs 0 /k , G =Tmo/0 (convective-cooling problem),

St = cs O' /9~, G =Tmo/0 ' (prescribed-heat-flux problem).

The non-dimensional melting temperature is given by

(9) Tm = G ( e x p [ - F ~ f ] - 1).

aT [ (prescribed-heat-flux problem). v y = k s - ~ = = o


The dimensionless energy equation can be written as follows:

(10) St Q s av 3~ -x a~ '

with the initial and boundary conditions specified as follows:

= o . ' ~ ~= 1

The condition at ~ = 0 is given by

aT 1 I I - T(O, ~) = 1

Bi ~ I ~ = o (convective cooling),

~T I = 1 (prescribed heat flux).

The non-dimensional energy is related to the temperature via

(11) T = ~U + fl,

where

$

r St ' ~1 J

~ = 0 , f l=Tm, {TmStr ~<b'<~l+TmStt)~} ' ~ ) 1

~ = % l c ~ S t ' ~ = ~ m ( 1 0~c~)01cl c~ l c l S t ' { b ' ~ > l + T m S t ~ } "

It can be observed that the temperature field inside the PCM depends on ~, v, St, F, G, Q1/Qs, X, cl/cs, To. In addition, the convective boundary conditions introduce a new parameter, the Biot number Bi.

3. - T h e n u m e r i c a l p r o c e d u r e

The energy equation has been approximated with the control volume finite- difference approach suggested by Patankar[20]. The resulting algebraic equations have been solved by the tridiagonal matrix algorithm.

The energy-density field U is first solved, using the temperature field given by the previous time step; then the solid-liquid interface is identified at the sharp discontinuity of the U-field. Through eq. (8) the PCM melting temperature is updated; the temperature field is corrected via eq. (7) and the problem is solved again. The procedure is repeated until convergence is attained. The converged results are assumed to be reached when the maximum relative change of the variables between consecutive iterations is less than 0.01%. This is a simple way to track with a trial and error technique the actual thermodynamic path followed by the PCM.


The consistency of the computational scheme has been checked by performing an overall energy balance at each time step: energy conservation was verified within 0.01% of the total heat removed from the PCM. However, the enLhalpy method applied to phase change problems can result in unaccurate temperature history and distribution inside the PCM, and careful attention must be paid in choosing the grid spacing as well as the time step. The check on the energy conservation is not effective to stress these difficulties, because it is basically tautologic with respect to the primitive equation. The entropy equation provides an alternative and independent way to verify the accuracy of the temperature field [21]. The entropy balance has been performed at each time step; the numerical source of entropy never exceeded 0.5% of the entropy generated inside the PCM.

4 . - R e s u l t s

The lack of experimental data makes it difficult to validate the present model. However, for low values of the Stefan number, the numerical solution has been tested against a quasi-steady approximation, developed for convective-cooling boundary conditions. When l inear temperature profiles are assumed in the solid and liquid phases, and as long as the PCM temperature at x = L0 is tied to To, the heat balance at the interface gives

d x f _ 1 I hk~ [Tm(xf)- T,~] k--I [To-Tin(x f)]}, (12) dt r hxf + k~ Lo - xf

where the actual melting temperature is coupled to the melting front position through eq. (6).

In terms of the non-dimensional variables, and for low F values, the heat balance becomes, after some manipulations

dr Bi ~f2 + (1 - Bi) ~f - 1 (13) St -

d~ kz _ Bi) Bi G(g -

Integration of eq. (13) gives, when kt/k~ = 1

_ _ [ A ( 1 - B i ) - F B i ] B i ~ f 2 F2Bi+FA(Bi -1 ) -A2 (in A ) (14) S t y = + A2 ~f+ 1+ ~ f 2A A 3 '

with A = Bi(1 + FG + 7"o) + FG and F - T0 - Bi. Figure 2 shows the melting front position vs. v, with F = 3.10 -2, St = 10-2;

curve a) refers to the quasi-steady solution and curve b) to the numerical solution. It can be observed that the agreement is quite satisfactory at low v values; when v becomes large the PCM temperature at x = L0 is driven towards Tm and the quasi-steady model fails down.

At large v values the results of the model can be checked by another technique. With convective-cooling boundary conditions, the melting front position saturates as


0.25 ~f 0.20

0.15

0.10

0.05

0.00 I I

2 4 ~ 6

Fig. 2. - The melting-front position vs. the non-dimensional time. Curve a) represents the quasi-steady solution, curve b) the present numerical solution for the convective-cooling problem. St = 0.01; F = 0.03; G = 100; T0 - 0; Bi = 150; r = 1.0905; Cl/Cs = 1.996; k l / k s = 1.

thermal equilibrium is attained. In this case equation (9) gives

(15) ~ q = 1 In G F G - 1

The numerical solution shows an excellent ag reement with this result: the percen tage discrepancy never exceeds 0.01%.

Now we are going to show, separately, the results for the two problems considered.

4"1. C o n v e c t i v e - c o o l i n g b o u n d a r y c o n d i t i o n s . - The influence of Bi, St, F, G, T0 on the melt ing f ront advancement will be analysed. The remaining non-dimensional pa r ame te r s have been chosen with reference to thermophysical proper t ies of water.

The effect of the Biot num ber is represen ted in fig. 3. He re St = 0.2, T0 = 1, F = 0.1 and G = 20. The melt ing front sa tura tes at ~f = 0.512. The curves show tha t for small Biot numbers a long t ime is required to at tain thermal equilibrium. A s t rong increase

1.0

0.8

0.6 |

Bi= lO

V , 0.0 5 10 15 20 ~ 25

Fig. 3. - The melting-front position vs. the non-dimensional time for different values of the Blot number. St = 0.2; F = 0.1; G = 20; To = 1; •t/Qs = 1.0905; cl/cs = 1.996; k l / k s = 0.271.


1.0

0.8

0.6

~ [ , , ~ S t= 0.1

0.0 5 10 15 I

20 ~ 25

Fig. 4. - The melting-front position vs . the non-dimensional time for different values of the Stefan number. F = 0.1; G = 20; To = 1; Bi = 10. Same notations as fig. 3.

1.0 ~f 0.8

0.6

0.4

0.2

~ F I 0"01

0.10

F = 0.30

o o 1'o 15 io ~ 2

Fig. 5. - The melting-front position v s . the non-dimensional t ime for different values of F. St = 0.2; G = 20; To = 1; Bi = 10. Same notations as fig. 3.

1.0 ~f 0.8

0.6

0.4

0.2

o.o ~ ~o i5 ~o �9 25

f

~ G = 3 0

Fig. 6. - The melting-front position vs . the non-dimensional time for different values of G. St = 0.2; F = 0.1; To = 1; Bi = 10. Same notations as fig. 3.


1.0

0.8

0.6

0.4

0.2

0.( 5 10 15 t

20 ~ 25

Fig. 7. - The melt ing-front position vs. the non-dimensional t ime for different values of the initial PCM temperature . St = 0.2; F = 0.1; G = 20; Bi = 10. Same notations as fig. 3.

1.0

if 0.8

0.6

0.4

0.2

I I

0.0 15 20 ~ 25

F= o O l - . . . / / / .

5 10

Fig. 8. - The me l t~g - f ron t position vs. the non-dimensional t ime for different values of F. St = 0.2; G = 8.58; To = 1. Same notat ions as fig. 3.

1.0

if 0.8

0.6

0.4

0.2

I I

0.0 15 20 T 25

G= 10 - - - - . . . -~ / / G= 20--- . . . . J~/ / G= 30

5 10

Fig. 9. - The melt-rag-front position vs. the non-dimensional t ime for different values of G. St = 0.2; F = 0.1; To = 1. Same notat ions as fig. 3.

268 M. CONTI, F. MARINOZZI and L�9 PAPAGNO

of the initial interface velocity is observed as Bi increases; indeed eq. (13) indicates that as ~f-* 0 the melting front velocity is given by ~(0) = St(Bi - Tokl/k~), i.e. it is proportional to the Biot number and is unaffected by the pressure-temperature coupling. At large Bi values the curves converge towards the solution for Dirichlet boundary conditions, with T(0, t) = Ta (curve d)).

Figure 4 represents the influence of the Stefan number. The curves show how faster thermal equilibrium is attained as the Stefan number increases. The effect of F, i.e. of the mechanical constriction on the PCM is shown in fig. 5. At F = 0.01 thermal equilibrium is attained when all the PCM is solidified. As F increases the velocity of the melting front is reduced and the liquid fraction of the PCM at thermal equilibrium increases, as indicated by eq. (15).

A similar role is played by G, as shown in fig. 6; eq. (9) indeed indicates that for low F values F and G affect the solution merely through their product.

Figure 7 shows the effect of the initial overheating of the PCM. The curves indicate that as T0 increases the adverse temperature gradient reduces the melting front velocity, but does not affect the saturation value ~f.

4"2. Prescribed-heat-flux boundary conditions�9 - Here the achievement of thermal equilibrium is disallowed by the continuous heat extraction at the surface x = 0. Simple considerations on the interface heat balance show that in the quasi-steady approximation the melting front velocity is given by

d~f 1 - To k~/k~ - ( 1 + FGkl ~ks) ~f (16) - St

dr 1 - ~f

Equation (16) shows that the pressure-temperature effects vanish as ~f--)0, where the interface motion is mainly affected by the Stefan number and by To. These suggestions are confirmed by the numerical results. Figure 8 shows the effect of different F values; due to the initial overheating of the PCM (To = 1) solidification starts at r > 0; the slopes of the curves collapse into a unique value as ~f--) 0. As ~ increases the effect of the mechanical constriction is evidenced: lower interface velocities correspond to higher F values, due to the growing adverse temperature gradient. The same behaviour can be observed in fig. 9, where the melting front position is represented at different values of G. The Stefan-number effect is shown in

1 . 0 m / . . . .

S t = 0 . 5 �9 ~ S t = 0 . 2

0 . 0 5 10 15 2 0 �9 2 5

Fig. 10. - The melting-front position vs. the non-dimensional time for different values of the Stefan number. F = 0.1; G = 8.58; To = 1. Same notations as fig. 3.

PRESSURE EFFECTS IN THE PLANAR SOLIDIFICATION ETC. 2 6 9

1.0

~f 0.8

0.6

0.4

0.2

z J r i

0.0 5 10 15 20 ~ 25

/

/

Fig. 11. - The melting-front position vs. the non-dimensional time for different values of the initial PCM temperature. St = 0.2; F = 0.1; G = 8.58. Same notations as fig. 3.

fig. 10; as predicted by eq. (16) the influence of St is reflected on both the short-time and the long-time behaviour of the interface motion.

Figure 11 shows the effects of the initial overheating of the PCM. As T0 increases the onset of the phase transition is retarded. The slope of the curves is strongly affected by T0 at the beginning of the process; at high ~f values the interface velocities tend to the same value. It means that the zero-order term in eq. (16) has become negligible in front of the first-order one, and the interface motion is mainly affected by the Stefan number and by the product F G k l / k s .

5. - C o n c l u s i o n s

In this paper we analysed the planar solidification in a slab of finite thickness; attention has been focused on the effects of the pressure increase due to the density change in the phase transition. Two different kinds of boundary conditions have been considered.

When the solidification is driven by convective cooling, the continuous advancement of the melting front is replaced by an asymptotic behaviour, until thermal equilibrium is attained.

When the boundary condition is specified in terms of a prescribed heat flow, the melting front velocity is slowed down by a growing adverse temperature gradient. The effects of the pressure-temperature coupling grow as the phase transition goes on, and vanish at the onset of the process. The influence of various parameters on the process has been presented and discussed.

R E F E R E N C E S

[1] CARSLAW H. S. and JAEGER J. C., Conduction of Heat in Solids (Clarendon, Oxford) 1959.

[2] PEDROSO R. I. and DOMOTO G. A., J. Heat Transfer, 95 (1973) 42. [3] CHARACH CH., ZARMI Y. and ZEMEL i . , J. AppL Phys., 62 (1987) 4375. [4] BIOT M. A. and DAUGHADAY H., J. Aerospace Sci., 29 (1962) 227.


[5] GOODMAN T. R., J. Heat Transfer, 83 (1961) 83. [6] VOLLER V. and CROSS M., Int. J. Heat Mass Transfer, 24 (1981) 545. [7] CRANK J. and GUPTA R. S., Int. J. Heat Mass Transfer, 18 (1975) 1101. [8] BONACINA C., COMINI G., FASANO A. and PRIMICERIO M., Int. J. Heat Mass Transfer, 16

(1973) 1825. [9] YAo L. S. and PRUSA J., Adv. Heat Transfer, 25 (1989) 1.

[10] LUNARDINI V. J., Heat Transfer with Freezing and Thawing (Elsevier, New York, N.Y.) 1991.

[11] ECKERT E. R. G. and DRAKE R. M., Analysis of Heat and Mass Transfer (McGraw-Hill, London) 1972.

[12] CRANK J., The Mathematics of Diffusion (Clarendon, Oxford) 1956. [13] IOM C.-J. and KAVIANY M., Int. J. Heat Mass Transfer, 33 (1990) 2721. [14] KIM C.~I. and KAVIANY M., Int. J. Heat Mass Transfer, 35 (1992) 457. [15] CHARACH CH. and RUBINSTEIN L, J. Appl. Phys., 71 (1992) 1128. [16] PATERSON W. S. B., The Physics of Glaciers, 2nd edition (Pergamon Press, Oxford) 1981,

p. 112. [17] HILLEL D., Applications of Soil Physics (Academic Press, New York, N.Y.) 1980, p. 257. [18] WEINGARTNER S. and BLUMENBERG J., Experimental and theoretical analysis of heat of

fusion storage for solar dynamic power systems, in XLI Congress of the International Astronautical Federation~ Dresden, Germany (October 1990).

[19] CONTI M., Int. J. Heat Mass Transfer, 38 (1995) 65. [20] PATANKAR S. V., Numerical Heat Transfer and Fluid Flow (Hemisphere, Washington,

D.C.) 1980. [21] BELLECCI C. and CONTI M., Solar Energy, 53 (1994) 163.

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