Pergamon Int. Comm. HeatMass Transfer, Vol. 23, No. 3, pp. 397--406, 1996

Copyright © 1996 Elsevier Science Ltd Printed in the USA. All rights reserved

0735-1933/96 $12.00 + .00

PII S0735-1933(96)00025-5

THE INWARD SOLIDIFICATION OF SPHERES WITH A SLIGHTLY PERTURBED TEMPERATURE DISTRIBUTION AT THE BOUNDARY

J Gammon and J A Howarth Dept of Applied Mathematics,

University of Hull, Hull HU6 7RX, UK.

(Communicated by P.J. Heggs)

ABSTRACT This paper is concerned with the inward solidification of liquid spheres where the liquid is initially at its fusion temperature and the temperature distribu- tion is axisymmetrically slightly perturbed at the boundary. Two different situations are considered i) where a constant temperature is maintained at the boundary and ii) where a constant heat flux is applied at the bound- ary. The problems are solved anaIytically by means of a large Stefan number approximation.

Introduction

In [1] Riley, Smith and Poots solved the problem of the sphere solidifying where a con-

stant, unperturbed temperature is maintained at the boundary and in [2] Howarth solved

the problem of the solidification of the sphere where there is a constant unperturbed heat

flux at the boundary. Here an analytic solution is presented for the problems where the

boundarytemperature or heat flux varies slightly from the constant value by an axisymmet-

ric perturbation function of the polar angle.

Analysis

Consider a spherical body, with coordinate r representing the radial distance from

the centre of the sphere. Initially the body is filled with liquid at its fusion temperature TF.

397

398 J. Gammon and J.A. Howarth Voh 23, No. 3

The boundary conditions at the wall where r = a are given by

either i) 7"* = T0(1 + e'f(O)) (1)

or ii)KO~-~7 - Q(l +ef(O)) (2)

where T* is the t empera ture of the solid state, I~ is the constant tempera ture mainta ined at

the boundary, Q is the magnitude of the heat flux applied at the boundary, e, e* are small

numbers, K is the thermal conductivity and f (0) is the per turbat ion function.

An axisymmetr ic solution is assumed, so o _~ 0 and the equation of conduction, when

expressed in spherical polar coordinates becomes

OT* = k { 1 0 (r2 OT* ~ 1 cO ( 0~_~_~ ) } cOt 7 ~ g \ Or ] + r2.~i~O O0 ~i~O (a)

subject to the boundary conditions

~* = 7~ .t r = a - S * ( o , t ) (4) (or" ar.os. dS"

I ( . ~ + r 2 00 0 0 / = - P L ~ i - at r = a - S * ( O , t ) (S)

S* = 0 when t = 0 (6)

where k is the thermal diffusivity, S* is tile radial distance of the solid/ l iquid interface from

the surface r = a, p is the solid density and L is tile latent heat of fusion.

Non-dimensionalise in the usual way by taking

r S* R = - , s = - - . (7)

a a

When there is a constant tempera ture at the boundary let

T* - To L kt Toe* T - - - f l - r - e -

TF -- To' c (Tr - To)' fla 2' TF - To

and when there is a constant heat flux at the boundary let

I~[ (T* -- TF) LI£ Qt T - Qa ' ~ - eQa' T = pLa

where c is the specific heat of the solid and fl is the Stefan number.

The problems become

lOT 1 0 (R2OT~ 1 0 ( ~ 0 ) fl & - R~ O R \ ~5~1 + R~ ~ino oo ,~i,~o

either i) T = ef(O) on R = l

(8)

(9)

(10)

(11)

Vol. 23, No. 3 INWARD SOLIDIFICATION OF SPHERES 399

O T

o--~ +

.. O T or zz)~-~ - (l + ef (0)) on R = I (12)

e i t h e r i ) T = 1 on R = 1 - S (13)

or i i ) T = 0 on R = I - S (14) 1 0 T O S OS

- o n R = 1 - S (15) R 2 00 00 Or

S = 0 on T=O. (16)

As in [3], we now seek perturbation functions of the form

T = To (R, S0) + eZl (R, 0, S0) + . . . (17)

S = S 0 -I- 6S1 (0, So) -~- • • • (18)

and use Taylor's theorem to apply the interface conditions at R = 1 - So rather than at

R = 1 - S. Note that, following [1] and [2] we use So, that is, the unperturbed interface

position, as our timelike variable, instead of r. (So is of course a monotonic function of r

only.)

The zeroth order problems are

l d S o O T o _ 1 0 tl" 20To\__~

d, OSo oR oR) (19)

e i t h e r i) To = 0 a t R = I (20)

or i i ) OT° O R - 1 a t R = I (21)

e i t h e r i )To = 1 a t R = I - So (22)

or i i )To = 0 at R = l - S 0 (23) OTo dSo

- a t R = 1 - S o (24) OR dT

So = 0 a t r = O (25)

with solutions

i) constant temperature (solved by Riley, Smith and Poots)

To= ( 1 - So) ( 1 - R) + ( 1 - R ) ( S o 2 - ( 1 - R ) 2 ) R S o 6t3 (1 - So) S g R

12/32( ~_--~)3RSo {~ ( 1 - ( T o R ) 2 ) + ( ~ ) ( 1 - ( ~ o R ) 4) } + . . .

sg (3 - 2So) sg sg r : + - - - + . . .

6 6,2 45/32 (1 - So) " z U ~ (~-)

So : U o ( T ) + - - = - - + + . . .

(26)

(27)

(2s)

400 J. Gammon and J.A. Howarth Vol. 23, No. 3

where

Uo = ~ + sin a r c s i n ( 1 2 7 - 1)

Uo U 1 -

6 ( U o - 1) Uo (1620o - 207)

u 2 = 3240 (Uo - 1 ) 3

and ii) constant heat flux (solved by Howarth)

1 1 1 { _/{2 1 1 T o - / { (1_ S o ~ + ~ 2(1 _ So)4 + 2 (1 _ So)2 + (1 _ S o ) ~

1 { - R 4 /{2 2I{ 1

q - ~ 30 (1 - So) z ÷ 3(1 ,5,o) s 3 (1 - So) 7 + 30(1 - So) 3

4 2 1 ( 2 +5(1- .5 'o) 8 3 ( 1 - S o ) 9 ÷ R 3(1-5 'o) s

1 /{ (1 - S 0 ) 4 }

1 + 3 (1 - S O ) 6

5 (1 So) 7

(29)

(3o)

(31)

(32)

1 1 (1 - So) 3 ÷ ÷ 7 - 3 3 ~ 6 3 (1 -So) 2

1 { 1 (1_So) 1 1 1 } +~T ~-~ 9 ( 1 _ o % ) 2 + 5 ( 1 _ , % ) 4 9(1 _ So)5

where

1

Uo = 1 - ( 1 - 3 T ) ~ u3 (Uo - 3 )

u1 - 6 ( 1 - U o ) 3

- 9U 3 + 54U~ - 135Uo 4 + 170Uo 3 - 40Uo 2 + 20Uo U2 =

180(1 - Uo) 7

when writing S0 in the form

The first-order perturbation

1 dSo OTI

fl dr OSo ei ther i) T1

or ii) OT1 OR OTo

T1 - & - - - OR

OT1 02To $ 1 - - 0/{ 0/{ 2

$1

of equation (28).

problems are

1 0 [R2OTI"~ I , 1 0 - / { 2 o/{ \ 3-ff] +/{2sino oo = f(O) at / { = 1

- - f (0) a t / { = 1

- - 0

= 0

a t R = 1 - So

dSo 0S1 at R = 1 - S o

dT 05o a t V = O.

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

Vol. 23, No. 3 INWARD SOLIDIFICATION OF SPHERES 401

Employing the method of separation of variables on equation (37) leads to a solution of the

form

T 1 = ~-~l~ln(R, So) P n ( # ) = ~ T l n (43) n = l n = l

Sa = ~ e ~ ( S 0 ) P~ (#) (44)

where p =cosO and P~ is the usual Legendre Polynomial. We may further assume that our

given surface or heat flux perturbation function may be decomposed as

f (0) = ~ F,~P,~ (#) (45) n = l

that is to say, the Fn may be regarded as given.

Note that we have disregarded the n = 0 mode, since this can be incorporated in the

unperturbed problem without loss of generality.

The equations become

R2 dsoO~ln 0 ( 0 ~ _ ) fl dT OSo OR R2 - t -n(n+ 1)gPln : 0 (46)

either i) ~ = F. at R = 1 (47)

or i i )02; ~ = - F ~ at R ~ - I (48)

OTo 0 1 ~ - e l n O R - 0 at R = I - S o (49)

001~ 02To dso Oel~ - a t R = 1 - S o (50) O---R - eln OR 2 dr OSO

el~ = 0 at r = 0 . (51)

The zeroth order problems are solved in [1] and [2] by means of a perturbation expansion for

large Stefan number, and this must therefore be mirrored in the solution of the first order

problem by writing

1 • ln = ~1o~ (n, So) + -)~Hn (n, So + . . . (52)

t

1 e~° = e~on(S0) + ~e, ln (So) + ' " . (53)

Substituting r (as a function of So), To and the expanded forms of Oln and el. into the equa-

tions gives the following:

402 J. Gammon and J.A. Howarth Vol. 23, No. 3

either (i) for the constant temperature case

O (e) O (1) terms:

0 ( O([DlOn ) n ( n + 1) (1DlO n -- ~ /~2 oR ) 0

(l)10 n = f;z at R = 1

~1o~+ elO~ - 0 at R = 1 - S o So (1 - So)

0I~D 10n 2elo~ delo~ So ( 1 - S o ) O R ( 1 - S o ) + d S ~ - 0 at R = I - S o

clO n = 0 at r = 0

O(e) O ( } ) terms:

O ( Oq'lX~ R 2 I2 (72 -~- 1) O l l n ~ R 2 : O(IDlon - ~ f i - / So (~ - So) OSo

4)u~ = 0 at R = 1

Clln C]On (I)11nq- So(1 - S o ) - 3 ( 1 - S o ) 2 S o at

0 (1D 11 n 2 e l l n d e l l n SO O(]) 1On S o ( i - S o ) OR ( 1 - So~) + dS~ - 3 OR

R = 1 - So

~10n

So (1 - So) at R = 1 - So

e l l n = 0 a l T = 0

or (ii) for the constant heat flux case

n ( n + l ) q h o ~ - ~ R 2

delon 2elOn

O(e) O(1) terms:

dSo (1 - So)

= 0

0(I) 10 n - I %

OR ¢lOn

4~1o,~+ ( 1 - S o ) 2 - 0 at

- - + ( 1 - So) 2 Oe)l°n OR - 0 at

elOn -- 0 a t

at R = 1

R = 1 - S o

R = 1 - S o

So=O

0@)0(~) terms:

(~ + 1) ~,~ - ~-~ R ~

8 R - 0

/{2 0(I)10 n

(1 - S o ) 2 0 S o

at R = 1

(54)

(55)

(56)

(57)

(58)

(59)

(60)

(61)

(62)

(63)

(64)

(65)

(66)

(67)

(68)

(69)

(70)

Vol. 23, No. 3 INWARD SOLIDIFICATION OF SPHERES 403

O l l n 4. e11n el0n ( 1 - - ( l -- S0)3) - - - a t R = l - S o (1 - So) 2 3 (1 - So) ~

0 ~ l l n 2elln delln /'1 _--_!1 ~So)3~ 0¢1o~ elOn (1 So) 2

OR ( 1 - S o ~ + d~-o - ~ 3 ( 1 - S o ) 2 J OR ( 1 - S o ) 2 at R = I - S o

(70

ella = 0 On S0.

(72)

(7a)

A somewhat lengthy, but essentially straightforward calculation, yields, after considerable

manipulation,

(I)10 n = AlOnR ~ + Blo,,R -(n+O (74)

(I)lln : A n n R ~ + B11~R -(n+l) + CunR n+2 4- DH~R l-n, (75)

where

(i) for the constant temperature case

(2~ + 1) (1 + 2r/) (1 - r/)~ Fn CIO n 6,1 (r/n+1 _ r/-~)

(~:10n + (1 -- r/) r / - n F n ) Alon =

(1 - r / ) (r /~+l _ r / - . ) elon + (1 - r/) r / n + l F n

~10n (1 - 7/) (r /n+l - - r/--n)

r/2n+3 r/1-2n

( r/~-~n (2~ + 1)(1 -- r/)~ + ~, (1~ C 2-n) + 6

~l ln :

Alln

- - + (2n + 1) 2

(2n ¥ ~ i ( i -- 2,~)) (2n + 1)

2(1 - 2n)

(76)

(77)

(78)

r/2) Fn}/(r /n+2--r /1-n)(79)

f elo,~ 1 dAlon ( r/~+2 r/'-n 2(2n + 1) r/-('~+')'~ "~ J

dAlon

dr/

/ ((1 - r/) @n+' - r/-n))

1 (2n + 1) Bun = - A n n + - -

(1 - q ) . ( 2 n + 3 ) ( 2 n - 1) 1 1 dAlo~

C l l n = 2 (2n + 3) (1 - r/)r/ dr/ 1 1 dAlon

D l l n = 2(2n - 1)(1 - 7/)7/ dr/

and (ii) for the constant heat flux case

(2n + 1) (1 - @) F~ el0 n = 3 ((n + 1) r/~+2 + nr/l-n)

((n + 1) el0 n 71- r/1-nFn) n l o n = ((,~ + 1) r/n+2 + , .71-n)

(80)

(81)

(82)

(8a)

(84)

(85)

404 J. Gammon and J.A. Howarth Vol. 23, No. 3

(n + 1)r/2~+a n2r/1-2n

(86)

- nr/2)

nAlon + Fn Blo~ -

(n + 1)

{ ( (2n+l)(2n3+an2-n-1) el ln = A l on 0-'~ .~_ ~) ~17 - - ~ (-~ 7 ~) (2n÷3)

{2n__+ 1 (n - 1)7/2 + n (n (2n + 1) - r/1-2~) +

\ at/ + 2 ( n + l ) ( 1 - 2 n )

/ ( (n + + nr/1- ) { ( n + l ) } { e,o~ (1 - r/a) e11,~

Alln : (n ÷ 1) r/n ÷ nr / - (n+l) 3r/6 7-12

dA1o,~ {" r/'~ (2n 3 + 3n 2 -- n -- 1) r/-O~+3)

÷ ~ ~,2(2n+3) ÷ (2n+a)(2n-1)(rz+1) 2 1 )

B]ln - (n + 1) dr/ r/2 (2n-+ 3~n7~ i •7 )n - 1)

1 dAlo,~ Cll n =

2 (2n + 3) r/2 dr/ n dAlo~

Olin : 2(n + 1) (2,~- 1)r/~ dr/

(1 - 2n) (n + 1)

-r/'-~Q Y~}

nr/-(n+') )} 2 ( n + 1 ) ( 2 n - 1)

(87)

(88)

(89)

(90)

(91)

where 7/= 1 - S0.

On combination of all modes, the final result will of course be

n=l

A symbolic manipulation computer programme was used to verify these results once they

had been obtained.

Results and Conclusions

Profiles of the inwardly-travelling front are displayed in Figure 1 and Figure 2, for the

constant t empera ture and constant heat flux perturbations respectively. For these examples

we have taken /3 = 10, e = 0.3. As an illustration, the perturbation function was simply

taken to be Ps(P) itself. Clearly more complicated functions can be obtained by simple

superposition.

It should be noted that the present analysis will not be valid close to final solidifi-

cation, for two reasons. Firstly, the unperturbed spherically symmetric analyses of [1] and

Vol. 23, No. 3 I N W A R D S O L I D I F I C A T I O N O F S P H E R E S 405

FIG 1 Profiles of the inwardly-travelling front where a

a constant temperature is maintained at the boundary.

FIG 2 Profiles of the inwardly-travelling front where a

a constant heat flux is mainta ined at the boundary.

406 J. Gammon and J.A. Howarth Vol. 23, No. 3

[2] themselves exhibit a s tructure requiring a matched asymptot ic expansions resolution very

near final solidification, and we have here per turbed only their outer solution, away from

that final small t ime interval. Secondly (and the reason for not a t tempt ing such a matched

asymptot ic analysis here at all) is that near the final solidification time, the perturbation

of the interface shape will be of the same order of magnitude as the unper turbed interface

radius, so that the region of unsolidified material may not, even be simply connected, and

the whole solution will cease even to be quali tat ively valid. A comprehensive analysis would

surely pose formidable difficulties, but one can expect that the solution presented here will

be valid away from the very end of the solidification process.

References

1. D S Riley, F T Smith and G Poots, The Inward Solidification of Spheres and Circular

Cylinders, Int J t teat Mass Transfer, 17, 1507-1516 (1974).

2. J A Howarth, Solidification of a Sphere with Constant tIeat Flux at the Boundary, Mech

Res Comm, 14, 135-140 (1987).

3. J A Howarth, The Stefan Problem for a Slightly Wavy Wall, Mech Res Comm, 17, 25-31

(1990).

Received August 11, 1995