LEXTERS IN HEATANDMASS TRANSFER Pergamon Press Vol. 5, pp. 19 - 28, 1978 Printed in Great Britain

SOLIDIFICATION OF A LIQUID PENETRATING INTO A CONVECTIVELY COOLED TUBE

Michael Epstein and George M. Hauser Reactor Analysis and Safety Division

Argonne National Laboratory Argonne, Illinois 60439

(Conrmanicated by J.P. Hartnett and W.J. Minkowycz)

Introduction

Solidification of a liquid at its fusion temperature as it penetrates

into an initially empty tube maintained at constant temperature was

treated theoretically and experimentally in [i]. An approximate method

was introduced which involves postulating a reasonable functional form

for the instantaneous shape of the frozen layer along the tube wall.

In [2] this approximate "crust profile method" was verified theoretically

using a numerical approach in which the governing integro-differential

equation of liquid motion was rigorously solved on a digital computer.

The good agreement obtained between the numerically exact solution and

the crust profile assumption in the special case of constant wall

temperature encourages exploitation~of the crust profile method to predict

the effects of a finite heat transfer coefficient when the flow tube is

immersed in a liquid coolant bath (see Fig. i). This system has been used

in previous experimental studies of freezing of an advancing flow [1,3].

Physical Model

With the exception that we here confine our attention to the frequently

encountered limiting case of negligible liquid flow inertia, our model

is identical with that of [i], viz. we consider a turbulent liquid flow

at its freezing temperature Tf penetrating into an initially empty tube

19

20 M. Epstein and G.M. Hauser Vol. 5, No.

T

7-%

z p i ~ X ~---

FIG. i

Solidification in flow into a tube immersed in a liquid coolant.

immersed in a liquid coolant (see Fig. i). As before [i], the assumption

of negligible axial heat conduction within the frozen layer implies

that the layer is thin compared with its extension in the direction

of flow and that the crust thickness is maximum at the channel entrance

(at z = 0). Clearly, the latter assumption is valid for a saturated

melt flow. In the region behind the advancing flow front (0 < z < X) the

;ube loses heat by convection at its outer wall at a rate ~ (heat flux),

which can be described by a power law; i.e.,

= h(T w - T=) n (i)

where h and n are constants, T is the initial temperature of the coolant

and the tube wall, and Tw(t ) is the unknown instantaneous temperature of

the tube wall. The quantity of interest is the flow penetration length

X at the instant solidification is complete at the tube inlet. P

Analysis

Assuming that the turbulent friction factor is given by the Blasius

Vol. 5, No. i SOLIDIFICATION OF A LIQUID 21

formula, the momentum balance between pressure and frictional forces

over the instantaneous penetration length X takes the form [i]

19j4 ] X 2 = 12.64 (AP Ro/ p)(2RoX/V)I/4 dz

_I

(2)

The crust shape R(z,t) must now be determined from the one-dimensional

conduction equation applied to the frozen layer. Unfortunately, no

exact solution for freezing a saturated liquid inside a cylinder is

available.

Stephan [4] adopted the following approximate method, originally

applied by Goodman [5] to problems of heat conduction with change of

phase:

(i) A reasonable functional form for the radial temperature profile

within the growing crust, T(r,t), is postulated, involving two undeter-

mined, time-dependent functions, viz., a "shape" function X(t) and the

radial location of the solid layer R(t):

r - T w in(r/Ro) [ in(r/R°) ] 2

Tf - Tw X in(R/Ro) + (I-x) in(R/Ro) (3)

(ii) The function X(t) is evaluated by imposing the condition that

T(r,t) satisfies a relation between (~T/~r)r= R and (~2T/~r2)r= R that is

explicitly independent of dR/dr and derivable from the energy balance

at the solid-llquid interface [viz., pLdR/dt = ks(~T/~r)r=R ] and the

conduction equation for the solid phase [viz., ~T/~t = (~s/r)~(r3T/~r)/~r]:

__ i ~ ~T Cs ~T + ~ ~ = 0 (4) L

r = r=R

This simple collocation technique yields the following ordinary differential

equation for R(t) calculated from the energy balance at the solid-liquid

interface:

I/2 dR [i + 2Cs(T f - Tw)/L] -i d-~ = ~ (5) s RIn(R/Ro)

22 M. Epstein and G.M. Hauser Vol. 5, No. I

The energy balance which equates the gain in wall sensible heat

to the sum of the conductive heat given up by the growing crust at the

inner tube wall and the heat lost to the coolant at the outer tube wall

is given by

dT (+r) w = -2=R k ~T -2~(R + 6t) ~ (6) PwCwAw dt o s r=R o

o

where A is the tube wall cross-sectional area and 6 is the tube thickness. + w t

It has been demonstrated in [1,2] that the axial variation of the

crust shape R(z,t) is well described by a function of the form

R 1 - (i -f)(l - z a ~-- = ~) ( 7 )

o o

where the exponent appearing in the above equation takes on values

in the range 0 < a < i. When a = 0 the solid phase is deposited

uniformly; i.e., R(z,t) = 6(t). We have a square-root solidification

profile and linear profile for a = 1/2 and a = i, respectively. The

profiles given by (7) span the entire range of physically possible

profiles, with a = 1/2 giving a solution that represents the exact

flow penetration history to better than 8 percent. Substituting (7)

into (2) leads to separable flow penetration velocity laws in terms of

the instantaneous radius at the entrance, 6(t). For example, for a = 1/2

we have

s4/7 d--~as =[8/15(A -15/4 -l)(l_A) z- 8/ii(&_ii/4_i ) ] -4/7 (8)

when cast in the dimensionless notation

1/3

s\ j 4B2~

s ; ~ ~ ~ t (9)

o

+The tube wall is treated as a perfect conductor of finite heat capacity.

Vol. 5, No. i SOLIDIFICATION OF A LIQUID 23

where

y z Cs(T f - T )/L ; i ] 1/2

B E + 2 7 -i ;

and the constant f is given by o

/64~30 ~ i/Ii f° = 0"0628 ~ o ~

Combining equations (3), (4), and (6), and introducing the additional

dimensionless quantities

Tw(t) - T 8 -

Tf - T

i OwCwAw (Ro + 6t)h(Tf - T~)n

A -= 2 PsCs(~R2 ) ; 8 -= ks(r f - r )

(10)

(11)

(12)

(13)

we obtain the ordinary differential equation

4AB dO i dT ln(A)

_i) ]_ oon (14)

Equations (8) and (14) can be solved along with the dimensionless form of

equation (5), viz,

d-~ = AIn(A--------) + 2y(l-e) -i

to determine the time history of the normalized location of the flow

front s(T), subject to the initial conditions s=0, A = 1 and 8 = 0 when

T = O.

(15)

Results and Discussion

For simple Newton cooling at the outer tube wall [n = i in equation

(1)] and negligible wall heat capacity (A+0), equations (8), (14),

and (15) reduce to the following quadrature form for the final penetration

length s : P

24 M. Epstein and G.M. Hauser Vol. 5, No. 1

11/7 44 ~ AF(A) s =--B [i ]I/2 p 7 Bln(A)-i + -B(I+2y)F(A)

1

] -417 8115(A-ISI4-1)-8111(A-1114-1) (16) • (l_A)Z dA

where F(A) = in(A) [2-BIn(A)] . Equation (16) is readily solved by the

Simpson rule and the results are displayed in Fig. 2.

For arbitrary n and A equations (8), (14), and (15) can be solved

by available computer subroutines that integrate systems of first order

ordinary differential equations. Since this more general case is seen

to involve a relatively large number of independent physical parameters

(viz., y,B, A, n), our purpose of illustrating a procedure is well served

by introducing parameter values appropriate to available experimental

data. Experimental penetration lengths have been reported in [1] for

water, benzene, and Freon II2A in copper tubes cooled by boiling saturated

liquid nitrogen. Initially, the liquid nitrogen pool and the empty

copper tube are in thermal equilibrium at the nitrogen saturation temp-

erature (T = - 196°C). At time t = 0, the hot liquid enters the copper

tube, the tube wall temperature T (t) begins to rise above T and boiling w

in the liquid nitrogen bath commences.

During the course of the experiment, the boiling liquid nitrogen

may pass through all the three boiling regions: nucleate, transition,

and film boiling. Merte and Clark [6] have obtained boiling heat-

transfer data with saturated liquid nitrogen using a transient technique.

Their study suggests the following piecewise discontinuous values for the

exponent n and the coefficient h in equation (i):

_3 h = 1.6 x i0 , n = 3.5; for -196°C < T < -187 (nucleate boiling)

w

h= 3.4

_2 h = 1.25 x i0 ,

n = 0.0; for -187 <

n = 0.71; for -167°C

T < -167 w

(maximum heat flux)

< T (film boiling) w

Vol. 5, No. 1 SOLIDIFICATION CF A LIQUID 25

L 3

i o - I i o 0 i o I

FIG. 2

Final penetration length for flow into a tube subject to a linear cooling law (n=l) on the outside and zero wall heat capacity (A=O).

_2 _i with ~ given in units of cal cm sec Three sizes of copper tubes

were employed in [i] having inside diameters 0.324, 0.476 and 0.631 cm

and wall thicknesses 0.0762, 0.0813 and 0.0813 cm, respectively.

The experimental penetration data reported in [i] are shown in

Fig. 3 along with the analytical predictions indicated by the dashed and

solid lines. + It can be concluded that the present simple model

exhibits the essential features of the penetration and freezing process.

It is significant to note that for the relatively low-conductivity Freon

ll2A, the present theory indicates that a constant tube-wall temperature

is maintained, i.e., nucleate boiling predominates in the liquid nitrogen.

The agreement with the theory for this material is seen to be good.

+The three theoretical curves for benzene correspond to the three different

tube diameters employed for this material.

26 M. Epstein and G.M. Hauser Vol. 5, No. i

o a-

x

1500

I000

500

0 0

I I ; I I i I

SYMBOL MATERIAL T.At.L(°C) D(cm)

• WATER - 196 0.476 o BENZENE - 196 0.476 WATER BENZE NE [] BENZENE -196 0.631 \ / / v BENZENE - 196 0.324 ' % . . . / / j , / • • • FREON II2A - 196 0.476 ~ . ~ . ~ • FREON 112A -196 0.324 / / ~ . ~ " ' - - ~ " j J _ • FREON ll2A 0 0 ' 4 7 6 v ~ - " = . ~ - ~ " • ~ ~ - c f ,. - ° 1 ~ [ ' j

. _ . [ /

• n n / ~ 1 _ _ ' ~ . l ~ ' / • FREON 112A

"~1 L I L I I J

n

I000 2000 3000 4000 5000 6000 7000 )-17/11

FIG. 3

Experimental penetration data [i] compared with approximate theoretical results.

Film boiling heat transfer limitations to benzene and water freezing

are predicted by the theory, and the corresponding penetration measurements

show less agreement with the numerical estimates, with the predictions

falling below the data by about 15 percent. This discrepancy may be

due, at least in part, to the fact that the freezing tube used in [i]

consists of a copper coil. Nitrogen bubbles formed at the lower coil

windings may blanket portions of the copper tube and thereby reduce

the boiling heat flux in the upper regions of the coil. The further

comment may also be made that equation (5) underpredicts the actual

Vol. 5, No. i SOLIDIFICATION OF A LIQUID 27

local channel radius-time relation.* We thus conclude that better

agreement between theory and experiment for all freezing materials could

be obtained by replacing equation (5) with a more elaborate solidification

model.

Acknowledgements

This work was performed under the auspices of the U. S. Department of Energy.

Nomenclature

a

A

A w

B

c

D

f o

h

k

L

n

P e

P o

Pr

r

R

R o

Re

crust profile exponent; equation (7)

dimensionless tube wall-crust heat capacity ratio; equation (13)

cross-sectional area of tube wall

solidification parameter; equation (i0)

heat capacity

tube diameter; Fig. I

d~mensionless friction parameter; equation (ii)

heat transfer coefficient; equation (I)

thermal conductivity

latent heat of fusion

exponent appearing in equation (I)

pressure at channel entrance; Fig. 1

initial pressure in empty channel; Fig. i

~/~, liquid Prandtl number

heat flux at outer tube wall; equation (i)

radial coordinate

distance from tube centerline to liquid-solid interface; Fig. i

D/2, channel radius i/2

(2AP/0) D/~, Reynolds number based on pressure drop AP.

* A comparison with available numerical solutions for the inward solid-

ification of cylindrical bodies [7] indicates that equation (5) constitutes

an accurate result for most situations of practical interest. In the

limit y + 0, equation (5) converges to the exact quasi-steady result;

it underpredicts solidification times by less than 20 percent for y < 2.0.

Unlikely physical situations characterized by y > 5.0 lead to somewhat

larger errors (>30 percent).

28 "M. Epstein and G.M. Hauser Vol. 5, No. 1

s

t

T

Tf

T W

T oo

X

X

Z

B

Y

t A

AP

O

P

%

X

dimensionless location of advancing liquid front; equation (9)

time

temperature within the growing crust; equation (3)

freezing point of liquid

temperature of tube wall

temperature of liquid coolant

location of advancing liquid front; Fig. i

velocity of advancing liquid front

axial coordinate measured from tube inlet; Fig. i

thermal diffusivity

Blot number; equation (13)

ratio of crust sensible heat to latent heat; equation (i0)

distance from tube centerline to liquid-solid interface at tube inlet; Fig. i

tube wall thickness

dimensionless location of liquid-solid interface at tube inlet

P - Po' pressure drop over the instantaneous flow penetratio~L length e

dimensionless tube wall temperature; equation (12)

kinematic viscosity

density

dimensionless time; equation (9)

temperature profile shape factor; equation (3)

Subscripts

p at completion of freezing

s solid phase

tube wall

References

i. M. Epstein, A. Yim, and F. B. Cheung, J. Heat Transfer, 99, 233 (1977).

M. Epstein and G. M. Hauser, J. Heat Transfer, 99, 687 (1977).

F. B. Cheung and L. Baker Jr., Nucl. Sci. Eng., 60, i (1976).

K. Stephan, Int. J. Heat Mass Transfer, 12, 199, (1969).

T. R. Goodman, ASME Trans., 80, 355 (1958).

H. Merte and J. A. Clark, J. Heat Transfer, 86, 351 (1964).

K. Stephan and B. Holzknecht, Waerme StoffUbertragung, i, 200 (1974).