Single correlation for theoretical contact melting results in various geometries

0735-1933/92 $5.00 +.00 Copyright © Pergamon Press Ltd.

INT. COMM. HEAT MASS TRANSFER Vol. 19, pp. 473-483, 1992 Printed in the United States

SINGLE CORRFJ.ATION FOR THEORETICAL CONTACT MEI JTING RESULTS IN VARIOUS GEOMETRIES Adrian Bejan J. A. Jones Professor of Mechanical Engineering Duke University Durham, North Carolina, 27706, USA

(Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT The melting rates due to close-contact heating of a block of phase-change material have been analyzed in the past based on thin-film lubrication theory in several internal and external configurations. The scale analysis of close-contact melting in a region of general shape shows that the melting rate in all configurations is anticipated by the expression

in which V is the speed with which the solid advances into the melting front (the melting rate), P/Ps is the liquid/solid density ratio, Ste is the Stefan number for liquid superheating (Ste << 1), and g and (z are the viscosity and the thermal diffusivity of the liquid phase. The film excess pressure scale AP is defined as the net weight of the object surrounded by liquid divided by the horizontal projected area of that object. The flow length scale of liquid film, £, stands for the diameter of the cylinder or the sphere, or for the smaller of the two sides of the rectangular shape of the contact area during melting against a flat beater. Review of Theoretical Results for Contact Meltin~ Theoretical results for close-contact melting have been reported for five basic geometries: 1) internal melting inside a horizontal cylindrical capsule, 2) internal melting inside a spherical capsule, 3) external melting around an embedded cylinder, 4) external melting around an embedded sphere, and 5) contact melting against a flat surface. The objective of this paper is to demonstrate that all these analytical results can be correlated, i.e. they can be anticipated (approximately) based on a single analytical expression. The single correlation developed in this paper is also theoretical (i.e. not empirical), as it is the result of the scale analysis of the thin-film melting and lubrication phenomenon. The melting of a phase-change material inside a horizontal cylinder (Fig. 1) was analyzed by Bareiss and Beer Ill. Initially the solid is at the melting point (Tin), and fills the cylinder. The wall temperature is raised to Tw at the time t = 0. The time tf needed to melt all the 473

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A. Bejan

VoL 19, No. 4

solid is given by the dimensionless expression ct tf = 2.491 p-- Ste1-3/4 (Pr Ar)-1/4 (1 + C)-1 Rz ~Ps 1

(1)

whea'e Pr ffiv / ~ and the Stefan and Archimedes numbers are defined by S t e - cp(Tw-Tm) hsf '

A r = ( 1 - p ~ ) gR3 V2

(2,3)

The term C is an empirical correction introduced to account for ac~idonal melting (a nfinor effect) associated with natural convection over the upper surface of the solid, C=0.25(~Ste

Ra 11/4 At/

(4)

in which l ~ = g~(T w - Tm)D3/(zv. Labeled s in Fig. ! is the distance traveled by the original

geometric center of the solid in the downward direction. This distance is the same as the largest liquid gap s(t) at the top of the solid, when melting over the upper surface of the solid is negligible. Bareiss and Beer [1] found that the solid falls with the nearly uniform speed, V = D/tf, because s(t) increases almost linearly from s(0) = 0 to s(tf) = D. Roy and Sengupta [2] and Bahrami and Wang [3] reported thin-film analyses for contact melting in a spherical enclosure. Roy and Sengupta's results are presented as a family of curves with Ste/Pr and Ar/(p/ps) as independent parameters. The predicted melting rate agrees well with

Li,

T

g

D = 2R

1 FIG. 1 Close-contact melting inside a capsule shaped as a sphere or horizontal cylinder.

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CONTACT MELTING IN VARIOUS GEOMETRIES

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Moore and Bayazitoglu's [4] experiments with n-octadecane. In addition, the analysis shows that the contact melting film is thinner at the lowest point of the spherical surface, and that the melting rate decreases as Ar/(p/ps) decreases. Bahrami and Wang [3], on the other hand, developed a closed form expression for the time interval needed to melt all the solid. If we use the Ar definition given in eq. (3), Bahrami and Wang's expression can be rewritten as ¢z tf = 2.03 If)

R2

~ss

Ste]-3/4 (Pr Ar)-1/4

(5)

I

The gradual fall of the solid is described by a curve s/D versus t/tf that is nearly the same as for melting inside a horizontal cylinder. In other words, the downward velocity of the solid is essentially constant in time, regardless of the shape of the capsule. If we turn the geometry of Fig. 1 inside out we arrive at Fig. 2, which shows how a hot object sinks into a larger body of solid phase-change material. Contact melting occurs over the leading portion of the hot object: the pressure built in the liquid film supports the weight of the object during its quasi-steady sinking motion. The liquid wake generated behind the object refreezes at some distance downstream ff the solid phase-change material is subcooled.

-

D=2R

,[

FIG. 2 External contact melting: heated sphere or horizontal cylinder sinking through a solid phase-change material.

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A. Bejan

eel. 19, No. 4

The earliest description of contact melting around embedded objects a ~ . a r s to have been reported by Nye [5]. His analysis dealt with unheated objects embedded in ice, where the melting is due to the pressure, i.e. the decrease of the melting point of ice as the pressure increases under the object. Emerman and Turcotte [6] studied the motion of a heated sphere (Tw) through a solid phase-change material whose temperature T,. is below its melting point (Tin). They found that the liquid film thickness 8(t~) must increase dramatically in the direction that points away from the "nose" of the sphere (0 = 0), 8(0) = a Ste , Vcos¢

(Ste << 1)

(6)

where the Stefan number ste = cp (Tw - Tin)

(7)

is based on the augmented latent heat of melting h~f = hsf + c s (Tm - Too)

(8)

The vertical velocity of the sphere, V, is prolx~'tional to the imposed temperature difference (Tw - Tin) raised to the power 3/4. If the Stefan number is much smaller than 1, the velocity is givcn by V R = Ste3/4

g a p R3.

(Ste << 1)

(9)

in which Ap is the differcn¢~ between the dunsity of the object (sphere) and the density of the surrounding melt, Ap = Po - P. The results summarized in cqs. (6)-(9) are valid only for small Stefan numbers. The more general results valid for any Ste value are listed in Emerman and Turcottc's [6] paper. The sinking of a horizontal cylinder embedded in a solid phase-change medium was studied experimentally and analytically by Moallemi and Viskanta [7,8], Their analytical results are valid for any Stefan number. In the Ste << 1 limit, the film thickness varies according to eq. (6) over the leading surface of the cylinder, while the vertical velocity V is given by VR =Ste3/4(1,,1~6_66" g a~--0c p R3)] 1/4'

(Ste<< 1)

(10)

The fifth basic geometry considered in this review is the contact melting of a block of phase-change material pressed against a flat heater (Fig. 3). This was studied analytically by

Vol. 19, No. 4

CONTACT MELTING IN VARIOUS GEOMETRIES

I Fn

477

Solid Phase-Change Material

_J ~

V

•

1

"-.

•

Liquid

"~'~:~~>~:~.:~.,'~: .* ~

~i~!:~,',.'~:~N~:~N:@~~:,,;~:,'~,~: ,'.,::~,~::~ .................................................................

.................................

Film

."4

[ -

I-

L

.....................~

-- [ I

[

Solid S~er

HG. 3 Block of phase-change material pressed against a plane slider. Moallemi et al. [9], Bejan [10], and Hiram et al. [11]. In Fig. 3, Bejan [10] further assumed that a) there is relative motion (U) between the melting solid and the hot slider, and b) the solid and liquid phases of the melting substance have almost the same density. He showed that the film thickness is independent of the speed U when viscous heating effects are negligible,

,'--

0)1 ,

I0,---0)

,,1>

In this equation FI is the nondimcnsionalized version of the average excess pressure experienced by the film, AP = Fn BL AP.L 2 ~ttz

rI = ~

(12)

(13)

The pressure drop number I I is an important dimensionless group that must be recognized in forced convection configurations in which the pressure difference is imposed [12]. The factor (~ accounts for the rectangular shape (ratio B/L) of the sliding contact area. The asymptotes of the ¢(B/L) curve are (~ ---) 1 when B/L >> 1, and 0 -') (B/L)2 when B/L << 1. The corresponding result for the melting rate is also independent of the slider speed [10],

(14) a

~ ~¢

l

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A. Bejan

VoL 19, No. 4

The contact melting resultsreviewed untilnow can bc anticipatedbased on a very simple analysis. W e write ~Lfor the longitudinallength scale of liquidflow through the film,and further assume that the contact surface is not necessarilyplane (Fig.4). The conservation of mass in the liquid~

requiresthat u8~ V g

(15)

The mon~ntum balance is simply AP

u ~ I.t 82

(16)

because, ff present, the shearing caused by relative motion (i.e. the Couette part of the film flow, e.g. Fig. 3) does not contribute at all to the longitudinal pressure gradient AP/t. FinaLly, the conse~,ation of energy at the melting front requires that k A T - Ps hsfV 8

(17)

If we eliminate u and 8 between eqs. (15)-(17), we obtain a melting speed that can be nondimensionalized in the form of the Peclet number based on t,

(18)

I Fn qolid

.~.Change tterial

FIG. 4 General shape of mating surfaces with close-contact melting.

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CONTACT MELTING IN VARIOUS GEOMETRIES

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Correlation of Existinf Theoretical Results The scaling law (18) reproduces exactly the plane contact melting result (14) if we set ffi L when L << B, and t ffi B when B << L, i.e. when Jt represents the shorter of the sides of the rectangular area of contact. More important is that eq. (18) also correlates the results for contact melting inside capsules and around embedded hot objects. First, we must recognize that the excess pressure scale AP can be defined as follows: AP = the net weight of the object surrounded by liquid the horizontal projected area of that object

(19)

In the case of melting inside capsules, the numerator in this definition represents the net initial weight of the solid phase-change material. The definition (19) yields AP ffi g Ap (~ D/4) for a horizontal cylinder, and AP = g Ap(2D/3) for a sphere. The excess density of the sinking object is Ap = Ps - P for melting inside a capsule (Fig. 1), and Ap = Po - P for objects embedded in a solid phase-change material (Fig. 2). The contact melting results reviewed earlier can be rewritten as follows:

Cylindricalcapsule,horizontal[Fig. 1, eq. (I) with (I + C) = 1, and V ffiD/if]:

-- -VD

1.015 P---Ste3/4

ps

(~P'~I ''4, (-GT-]

(.q. = 0.971D)

(20)

(Jr ffi 0.595D)

(21)

Spherical capsule [Fig. 1, eq. (5) with V = D/tf]: VD

ffi

1.297 P---Ste3/4

(~d:~"D2/114 '

Embedded horizontalcylinder[Fig.2, eq. (I0)]: lAP. D 2~1/4 ) '

VD cx = 1"257 Ste3/4 t ~ ' -

(.o. = 0.633 D)

(22)

(t=O.353D)

(23)

Embedded sphere [Fig. 2, eCl. (9)]:

/'~LP D 2\1/4 Y' D = 1 " c6 8 2 S t eo3 / 4 t ~ - ~ )

Equations (14) and (20)-(23) show that the scaling law (18) anticipates within percentage points the melting speed in all geometries, if the length scale Jt is interpreted as the actual dimension of the projected area of contact, namely,

480

A. Bejan

..o..= D, in Figs. 1 and 2

Vol. 19, No. 4

(24)

and = rain (L,B), in Fig. 3

(25)

There is some disagreement with regard to the role played by the density ratio PIPs. Note that this ratio enters as (p/ps)3/4 in the scaling law (18), as P/Ps in the formulas (20,21) for melting inside capsules, while being absent from the results (22,23) for melting around embedded hot objects. Numerically, however, the effect of P/Ps is very small, because this ratio is a number close to 1. Listed in parentheses to the right of eqs. (20)-(23) are the ~ values that would make these equations agree exactly with eq. (18), again, if we discount the small piPs effect. In this case, "exact agreement" means that the leading numerical factor on the right-hand side of eqs. (20)-(23) is equal to 1. This last observation suggests that the correlation (18) would work even better if the film length scale ~ is chosen based on a nile that accounts for the shape and curvature of the contact melting surface. For example, g may be defined as the ratio of the contact melting area A divided by the perimeter of that area projected on the plane perpendicular to the normal force, p (e.g. Goldstein et al. [13]) ~-0 = A

(26)

P For cylinders and spheres, A is approximated weU by the leading haft of the total surface of the object (e.g. Fig. 2), so that ~0 = (x/4)D for the half-cylinder, and g0 = D/2 for the hemisphere. The corresponding length scale of the plane contact area L x B is ~0 = (1/2) min (L,B). If we employ the length scale ~-0 instead of the ~. scale of eqs. (24,25), the five theoretical results reviewed in eqs. (14) and (20)-(23) become, respectively: Plane rectangular heater a

(14')

~go~]

Cylindrical capsule -

p

=

ps

~ ~oc J

(20')

Spherical capsule V .0.o ffi 0.92 ~ Ste3/4/{&p.~2x|1/4 Gt Ps ~ laa !

(21')

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CONTACT MELTING IN VARIOUS GEOMETRIES

481

Embedded cylinder [

•

,, 2

V~to=a 1.11 St¢3/4 t ~ - ~

~1/4

)

(22')

Embedded sphere V to = 1.19 Ste3/4 (AP" -°-o2)t/4

C2a--)

O~

(23')

Equations (14') and (20')-(23') show that if the film length scale is chosen based on the rule (26),the scaling law (18) anticipatesmore accurately the five theoreticalresultsthat we have been comparing. Note that especially in exls.(20')-(23')the numerical factor on the right-hand side is approximately the same as the factor I used by default in the proposed correlation(18). Notation

A Ar B cp C D Fn g hsf h~ k

area of contact melting [m2] Archimedes number, cq. (3) length perpendicular to Fig. 3 [m] liquid specific heat at constant pressure [J/kg.K] natural convection correction, cq. (4) diameter of sphere or cylinder [m] normal force [N] gravitational acceleration [m/s2] latent heat of inciting [J/kg] augmented latent heat of melting, eq. (7) [J/kg] liquid thermal conductivity [W/m.K]

J~ ~-0

length scale of thin film [in] film length scale,cq. (26), [m]

L m m0 p

swept length, Fig. 3 [m] instantaneous mass of solid, Fig. 1 [kg] initial mass of solid, Fig. 1 [kg] projected perimeter of contact melting area [m]

Pr

Prandtlnumber, v/co

R Ra

radius of sphere or cylinder [m] Rayleigh number, cq. (4)

s Stc

downward travel of solid,Fig. 1 [m] Stcfan number, cq. (2)

tf Tm

duration of melting process inside capsule Is] molting point [K]

Tw

wall mn~mmre

[K]

482 T~ u U V V x,y a J3 8 AP AT Ap Ix v H p Po Ps ¢

A. Bejan

VoL 19, No. 4

temperature of solid phase-change medium, Fig. 2 [K] liquid velocity component in the longitudinal direction [m/s] slider velocity, Fig. 3 [m/s] vertical velocity of embedded object, Fig. 2, or melting block, Fig. 3 [m/s] averageverticalvelocityof solidmelting in capsule,Fig. I,V ffiD/if[m/s] cartesiancoordinates,Fig. 3 [m] liquidthermal diffusivity[m21s] volumetric coefficient of thermal expansion [K-l] liquidfilm thickness[m] excesspressure,eq. (12) [N/m 2] temperaturedifference,Tw - Tm, [K] densitydifference,Po - P, or Ps - P, [kg/m 3] viscosity [kg/s.m] kinematic viscosity [m2/s] excess pressure number, eq. (13) liquiddensity [kg/m 3] densityof embedded object,Fig. 2 [kg/m3] solid density [kg/m3] angular coordinate, Fig. 2 geometric factor, ¢ = ~(B/L) References

1.

M. Bareiss and H. Beer, An Analytical Solution of the Heat Transfer Process During Melting of an Unfixed Solid Phase Change Material Inside a Horizontal Tube, International Journal of Heat and Mass Transfer 27, 739-746 (1984).

2.

S.K. Roy and S. Sengupta, An Analysis of the Melting Process Within a Spherical Enclosure, ASME Vol. SED 1, 27-32 (1985).

3.

P.A. Bahrami and T. G. Wang, Analysis of Gravity and Conduction-Driven Melting in a Sphere, Journal of Heat Transfer 109, 806-809 (1987).

4.

F.E. Moore and Y. Bayazitoglu, Melting Within a Spherical Enclosure, Journal of Heat Transfer 104, 19-23 (1982).

5.

J.F. Nye, Theory of Regelation, Philosophical Magazine 16, 1249-1266 (1967).

6.

S.H. Emerman and D. L. Turcotte, Stokes's Pproblem with Melting, International Journal of Heat and Mass Transfer 26, 1625-1630 (1983).

7.

M.K. Moallemi and R. Viskantg Malting Around a Migrating Heat Source, Journal of Heat Transfer 107, 451-458 (1985).

8.

M . K . Moallemi and R. Viskanta, Experiments on Fluid Flow Induced by Melting Around a Migrating Heat Source, Journal of Fluid Mechanics 1~7, 35-51 (1985).

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483

.

M. IL Moallemi, B. W. Webb and R. Viskant& An Experimental and Analytical Study of Close-Contact Melting, Journal of Heat Transfer 108, 894-899 (1986).

10.

A. Bejan, The Fundamentals of Sliding Contact Melting and Friction, Journal of Heat Transfer 111, 13-20 (1989).

11.

T. Hirala, Y. Makino, and Y. Kaneko, Analysis of Close-Contact Melting for Octadecane and Ice Inside Isothermally Heated Horizontal Rectangular Capsule, International Journal of Heat and Mass Transfer 34, 3097-3106 (1991).

12.

A. Bejan, Heat Transfer, Wiley, New York, Chapter 6 (1993).

13.

R. J. Goldstein, E. M. Sparrow and D. C. Jones, Natural Convection Mass Transfer Adjacent to Horizontal Plates, International Journal of Heat and Mass Transfer 16, 1025-1035 (1973).

Received April 20, 1992