Solidification in a semi-infinite region with boundary conditions of the second kind: An exact solution

IN}~ATAbD~S TRANSFER 0094-4548/79/0701-0321502.00/0 Vol. 6, pp. 321-327, 1979 © P e r g a m c n P r e s s Ltd. Printed in Great Britain

SOLIDIFICATION IN A SEMI-INFINITE REGION WITH BOUNDARY CONDITIONS OF THE SECOND KIND: AN EXACT SOLUTION

Mohamed S. E1-Genk EG&G Idaho~ Inc. P.O. Box 1625 Idaho Falls9 Idaho 83401

August W. Cronenberg Dept. of Chemical and Nuclear Engineering The University of New Mexico Albuquerque, New Mexico 87131 (C~t,~nicated by J.P. Hartnett and W.J. M i n k ~ c z )

ABSTRACT This paper presents an exact analytlcal solution for the freezing of a stagnant liquid in a seml-lnfinlte slab geometry subject to a constant heat flux boundary condition, commonly referred to as a boundary condition of the second kind. Both the transient solid-liquid interface position and temperature field in the solidifying layer are obtained. Introduction The exact analytical solution to most problems of transient heat conduction with a moving phase-transformation boundary is not readily obtainable because of the nonlinearity of the boundary condition at the moving interface.

The only known exact solutions to transient freezing or melting

problems are those for~

a seml-lnflnlte slab with a boundary condition of

the first kind (constant temperature), conmlonly referred to as the Neumann problem Ill ; a finite slab subjected to a specific heat flux input, assuming that all molten material is removed immediately upon formation[2]; and the freezing of a flowing liquid on an isothermal cold wall[3]. With respect to the problem at hand [that is, freezing (or melting) with a boundary condition of a second kind] Evans, et al[4] have presented a closed series solution for the melting in a semi-infinlte slab at its fusion temperature, assuming a power series in time for the instantaneous position of the change-of-phase front, 6(t), and a double power series in time and 321

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Vol. 6, No. 4

space f~r the temperature distribution in the molten layer.

The solution to

this problem[4] entails tedious mathematical manipulations in addition to being unbounded for times t > 0.25

/qc

~s

as will be shown later in this work. An approximate analytic solution to this problem[~has also been introduced by Goodman [9 using the integral heat balance technique, in which an implicit expression for the moving interface ~(t) is obtained as a function of 6(t) and time.

freezing (or melting) problem

The exact solution to the one-dimensional

of a saturated liquid in semi-infinite slab geometry subject to a constant heat flux is found and presented in the present paper.

The general tech-

nique is similar to that for obtaining a solution to a nonlinear partial differential equation; that is 9 assuming the solution and then checking it for the governing equation and both the boundary and initial conditions. Table 1 presents a comparison of nonlinear

freezing (or melting) problems

with similar linear transient heat conduction problems.

Fig. I illus-

trates the physical model of the present problem. Analysis A warm liquid is considered in the half space x > 0 which is at a uniform temperature equal to its fusion temperature Tf and subject to a constant heat flux qc at its surface 9 x = 0.

At time zero9 a solidifi-

cation front 6(t) appears at the wall surface, x = 09 and propagates in the positive x-direction into the liquid.

The temperature field in the frozen

layer, shown in Fig. I, should satisfy the transient heat conduction equation ~T s

=

a

~t

~2T ----E s. s ~x 2

(1)

The boundary conditions at the wall is of constant heat flux qc

Ks ~~x Ix=0

=

qc"

(2)

The temperature of the frozen material at the moving front, x = 6(t), is equal to the fusion temperature Tf Ts(6,t)

=

Tf.

(3)

condition

Complete solution

~enerel fern of solution

Inltiet

Bouudaz~7 con8 i t i o n s

C~veru/ng eqont ion

It=~,, as

az 2

~2Ts

0sL d6

erf

x

x

(-,t)

~r 9/ = 0

Ts(x,t) - r o

÷ (T

~ - T o) e r r

the same

x

(conzteet)

rz(x,0) " r.

" T 0eL d6 g s dt

÷ B(t)

terfc

ier~

2 x

x

where 6 ( t ) is d e t e r m i n e d from • q. (9)

erfc 26t

Ts(z,t)=A

6(t - o) - o

~rs ( 6 , t ) ax

Ts ( 6 , t ) - r f

(-,t)

" 0

(constant1

erfc

~

the same

e

Te(x,O) - ~ .

~T ~/

Ts (-~t) - T

the same

~r e Ks - ~ z ( O , t ) - qc ( c o n s t a n t )

the s a l e r e (-,t)

t h e same

Similar transient ~eat' c o n d u c t i o n problem [6J

t h e same

Freezing (or mett£ng) ( p r e s e n t wor k }

With boundary c o n d i t i o n x f t h e second kind (constant h e a t f l u x )

the name

S i m i l a r t r a n s i e n t ~e~t c o n d u c t i o n problem L1J

kind

2/~st

2/~t

end ~ is t h e f r e e z i n g (or ~ l t i n 8 ) constant Vhich can be dete~nixed b' s o l v i n g the t r z n s e n d e n t s l e q u a t i o n _X 2 (e l e r f X ) - ( ~ L , ~ C p s T f )

~(t) - 2 /~.t

where

T s ( X , t ) " Tx + ( T f - Tx) e r r

+ D err

6 ( t - o) - o

rs(z,t)=c

~Ts

r e ( 6 , t ) - Tf

T z ( O , t ) = TO ( c o n s t a n t )

~t

~Tm

F r e e z i n g ( o r mel~i~g) (lqeumsn FroblemLlJ)

VLth boundsry ¢ o n d £ t l o n x f the f i r s t (constant temperature)

A Comparison o f Nonlinear F r e e z i n g ( o r M e l t i n g ) Problems w i t h the S i m i l a r L i n e a r T r m l s i e n t Heat Conduction Problems

TABI~ I

~0 ~O

l

d~

O~

F~

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1

I

~

I I IIII

Vol. 6, No. 4

I

I

!

I I!11

-I

E

i }

=

=. ~

J

[

J

Exact ~olul~n

~ . ~

Evans et m end the

|

~ul,ons

I I I l i l

~0-2

I

"~

_ . . I

[IIIL

!

10 -1 Oa~stonN~s

t~me

r

INEt= A 10 77~

FIG. 1 Physical Model for the Freezing of a Stagnant Saturated Liquid in a SemiInfinite Slab Subject to a Constant Heat Flux.

The energy balance at the moving interface gives

aTs I

Ks~

~6(t)

=

d6

(4)

~s L~-~.

The solution for the temperature field in the solidified crust Ts(X,t) is assumed to be of the form

Ts

(x,t)

=

A + B(t)

"

ierfc (x/2 avast).

0 ~ x E 6(t)

(5)

It is to be noted that Eq. (5) is the general solution for the transient heat conduction problem in a semi-infinlte slab subject to a constant heat flux boundary condition without change-of-phase[6], Table I.

The time-dependent

as demonstrated in

coefficient B(t) and the constant A in Eq. (5)

are determined by imposing the conditions that Ts(X~t) satisfies Eqs. (1)-(3).

An expression for the position of the change-of-phase front ~(t)

is then obtained by applying the energy balance condition given by Eq. (4). It can be seen that Eq. (i) is already satisfied. The boundary condition given by Eq. (2) requires that

S(t)

and

from

Eq.

=

-

K2qc d~ssst' s

(6)

(3) that

A

=

2q c Tf + ~--- avr~-st [ i e r f c S

(6/2 sv~-) ] .

(7)

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SOLIDIFICATION IN A SEMI-INFINITE R~GION

325

Therefore, Eq. (5) becomes 2q c

Ts(x,t)

=

+ r-

[ierfc

-

(8)

ierfc

S

Now, the instantaneous position for the change-of-phase front 6(t) is determined from the remaining boundary condition [Eq. (4)]9 using Eq. (8), such that

d--t = d6

(9)

PsL erfc 1 2 ~ s t ) qc

When the following dimensionless parameters are introduced, that is =

(dimensionless frozen layer thickness)

6(t),

and

T

qc p2 L

= S

t, (dimensionless time),

(I0)

S

Eq. (9) becomes d~

d-V

A

=

erfc 2 J

(I1)

The solution for A(T) from Eq. (Ii) is easily obtained numerically, using the error function tables making use of the fact that the initial condition in the frozen layer is

A(t = O) = O.

(12)

Eq. (II) gives the freezing (or melting) velocity for a saturated liquid on a cold wall, subject to a constant heat flux boundary condition. For such a problem, Evans, et al[4] presented the following expression for molten layer thickness A(T), using a series solution technique, such that

A(T)

is obtained as A(T)

1 ffi z - ~-T

T2

+

~.~3 - 4-T514

+

~827T5

- " -"

(13)

An implicit expression for the molten layer thickness ~(T) is given by Goodman[5], using the integral heat balance method~ as A(T)

=

A(T) [A(~)+ 5 +

~

(1 + 4

A(T))I/2 l •

(14)

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Vol. 6, No. 4

Results and Discussion The solution presented is the first known exact solution to the

freezing (or melting) problem of a stagnant liquid at its fusion temperature in a semi-infinite slab subject to a constant heat flux boundary condition. A plot of the dimensionless frozen crust, A(r), versus the dimensionless time, z, is presented in Fig. 2, where the accuracy of both the series solution of Evans et al[4]and the integral solution of Goodman[5] is also investigated.

As indicated, the series solution[4] is unbounded for the

dimensionless time z > 0.25.

The integral solution[5] predicts the exact

frozen crust thickness to better than 8.5%; however, the solution presented here has the advantage of being both exact and readily obtainable.

Frozen crust Stagnant saturated liquid Tf

Ts(x t)

Constant heat tlux

6 (I)-,~,D,

m~

g

INEL-A 10 777

0

FIG. 2 The Transient Dimensionless Frozen Layer Thickness for a Saturated Liquid Undergoing Freezing.

Notation A

Arbitrary constant, Eq. (5)

B

Arbitrary coefficient, Eq. (5)

C

P

Specific heat

K

Thermal conductivity

L

Latent heat of fusion

qc

Constant heat flux at the wall-frozen crust interface

t

Time

T

Temperature

x

Distance measured from the wall-frozen crust interface

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SOLIDIFICATION IN ASEMI-INFINITEREGICN

Greek Letters Thermal diffusivity 6

The instantaneous

Dimensionless 0

Density

T

Dimensionless

frozen crust thickness

frozen crust thickness, Eq. (I0)

time, Eq. (I0)

Subscripts f

Fusion

s

Solidified crust References

I.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed., Chapter 11, Clarendon Press, Oxford, 1959.

2.

R. W. Sanders, ARS J, Vol. 30, 1960, pp 1030.

3.

P. bl. Heertjes and Ir. Ong Tjing Gie, British Chem. En~r. , Vol. 5, 1960, pp 413-419.

4.

C. W. Evans, E. Isaacson, and J. K. L. MacDonald, Quarterly of App1. Math., Vol. 8, 1950, pp 312-319.

5.

T. R. Goodman, J. Heat Transfer, Vol. 80, 1958, pp 335-343.

6.

A. V. Luikov, Analytical Heat Diffusio6 Theory, 2nd ed., Chapter 5, Academic Press, Inc., New York, 1968.

327