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

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

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

474 A. Beja n 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 R z ~Ps 1

whea'e Pr ffi v / ~ and the Stefan and Archimedes numbers are defined by

S t e - - cp(Tw-Tm) A r = ( 1 - p ~ ) gR3 hs f ' V 2

(1)

(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 . 2 5 ( ~ S t e 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,

g T

D = 2R

1 FIG. 1

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

VoL 19, No. 4 CONTACT MELTING IN VARIOUS GEOMETRIES 475

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

If) Ste] -3/4 (Pr Ar) -1/4 ¢z tf = 2.03 (5) R 2 ~ss 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 = 2 R ,[

FIG. 2

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

476 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 , (Ste << 1) (6) Vcos¢

where the Stefan number ste = cp (Tw - Tin) (7)

is based on the augmented latent heat of melting

h~f = hsf + c s (T m - 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

VR = 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 p R3) 1/4 " ~--0c ] ' (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 477

• 1 ~ V

I Fn Solid Phase-Change Material

_J

"-. • L i q u i d Film

"~'~:~ ~>~:~. :~.,'. ~: * ~ ~i~! :~,',.'~:~N~:~N:@~ ~: ,,;~ :,'~ ,~: ,'.,: :~,~:: ~

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ."4 ................................................................. .....................~ [

[ - - [ Solid S~er I - L - I

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 (12) B L

AP.L 2 r I = ~ ( 1 3 )

~ttz

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],

a ~ ~ ¢ l (14)

478 A. Bejan VoL 19, No. 4

The contact melting results reviewed until now can bc anticipated based on a very simple

analysis. We write ~L for the longitudinal length scale of liquid flow through the film, and further

assume that the contact surface is not necessarily plane (Fig. 4). The conservation of mass in the

liquid ~ requires that

u 8 ~ 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 (17) 8

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.

VoL 19, No. 4 CONTACT MELTING IN VARIOUS GEOMETRIES 479

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 (19) the horizontal projected area of that object

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:

Cylindrical capsule, horizontal [Fig. 1, eq. (I) with (I + C) = 1, and V ffi D/if]:

-- -- (~P'~I ''4, VD 1.015 P---Ste3/4 (.q. = 0.971D) (20)

ps (-GT-]

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

ffi (~d:~" D2/114 ' VD 1.297 P---Ste3/4 (Jr ffi 0.595D) (21)

Embedded horizontal cylinder [Fig. 2, eq. (I0)]:

lAP. D 2~1/4 VD = 1"257 Ste3/4 t ~ ' - ) cx ' (.o. = 0.633 D) (22)

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

/'~LP D 2\1/4 Y D = 1 " 6 8 2 S t e 3 / 4 t ~ - ~ ) ' c o ( t=O.353D) (23)

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 Vol. 19, No. 4

and

..o.. = D, in Figs. 1 and 2 (24)

= 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 ~ g o ~ ]

Cylindrical capsule

Spherical capsule

- p

= ps ~ ~oc J

ffi 0.92 ~ Ste3/4/{&p. ~2x|1/4 Ps ~ laa !

(14')

(20')

V .0.o (21') Gt

Vol. 19, No. 4 CONTACT MELTING IN VARIOUS GEOMETRIES 481

Embedded cylinder

Embedded sphere

[ • ,, 2 ~1/4 V~to=a 1.11 St¢3/4 t ~ - ~ ) (22')

= 1.19 Ste3/4 (AP" -°-o2) t/4 C2a--) V to (23') O~

Equations (14') and (20')-(23') show that if the film length scale is chosen based on the

rule (26), the scaling law (18) anticipates more accurately the five theoretical results that 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 area of contact melting [m 2]

Ar Archimedes number, cq. (3)

B length perpendicular to Fig. 3 [m] cp liquid specific heat at constant pressure [J/kg.K] C natural convection correction, cq. (4)

D diameter of sphere or cylinder [m]

Fn normal force [N] g gravitational acceleration [m/s 2]

hsf latent heat of inciting [J/kg] h~ augmented latent heat of melting, eq. (7) [J/kg] k liquid thermal conductivity [W/m.K] J~ length scale of thin film [in]

~-0 film length scale, cq. (26), [m]

L swept length, Fig. 3 [m]

m instantaneous mass of solid, Fig. 1 [kg]

m0 initial mass of solid, Fig. 1 [kg]

p projected perimeter of contact melting area [m]

Pr Prandtl number, v/co

R radius of sphere or cylinder [m]

Ra Rayleigh number, cq. (4) s downward travel of solid, Fig. 1 [m]

Stc Stcfan number, cq. (2) tf duration of melting process inside capsule Is] Tm molting point [K] Tw wall mn~mmre [K]

482 A. Bejan VoL 19, No. 4

T~ temperature of solid phase-change medium, Fig. 2 [K]

u liquid velocity component in the longitudinal direction [m/s]

U slider velocity, Fig. 3 [m/s]

V vertical velocity of embedded object, Fig. 2, or melting block, Fig. 3 [m/s]

V average vertical velocity of solid melting in capsule, Fig. I, V ffi D/if [m/s]

x,y cartesian coordinates, Fig. 3 [m]

a liquid thermal diffusivity [m21s]

J3 volumetric coefficient of thermal expansion [K -l]

8 liquid film thickness [m]

AP excess pressure, eq. (12) [N/m 2]

AT temperature difference, Tw - Tm, [K]

Ap density difference, Po - P, or Ps - P, [kg/m 3]

Ix viscosity [kg/s.m]

v kinematic viscosity [m2/s]

H excess pressure number, eq. (13)

p liquid density [kg/m 3]

Po density of embedded object, Fig. 2 [kg/m3]

Ps 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).

Vol. 19, No. 4 CONTACT MELTING IN VARIOUS GEOMETRIES 483

.

10.

11.

12.

13.

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).

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

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).

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

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