Experimental inward solidification of initially superheated water in a cylinder

INT. CIlMM. HEAT MASS TRANSFER 0735-1933/87 $3.00 + .00 Vol. 14, pp. 21-31, 1987 ©Pergamon Journals Ltd. Printed i n t h e U n i t e d S t a t e s

EXPERIMENTAL INWARD SOLIDIFICATION OF INITIALLY SUPERHEATED WATER IN A CYLINDER

W.E. Stewart, Jr. and K.L. Smith Energy Research Laboratory Department of Mechanical and Aerospace Engineering University of Missouri-Kansas City Truman Campus Independence, Missouri 64050

(Ccmmunicated by J.p. Hartr~tt and W.J. Minkowycz)

ABSTRACT The subject of this investigation was the freezing of initially superheated pure water, at small Stefan numbers, contained in a horizontal cylinder. Three experiments were conducted and were compared to an analytical model based upon the heat balance integral method which considered one-dimensional (radial) conductive heat transfer with either zero or a finite amount of initial superheat contained in the water. The equations are solved numerically, employing the Runge-Kutta method for solving the first order governing differential equations. The solution yielded the radius of the phase change boundary as a function of time and the time for the liquid to reach the phase change temperature (0°C) when there is initial superheat. The analytical and experimental solidification time results obtained in this investigation compare very well. As in previous studies, the solidification time was found to be a linear function of Stefan number for zero initial superheat. The analytical results obtained for no initial superheat, though, differ somewhat from the results of some other investigations. Free convection affects appeared to be negligible.

Introduction One of the experimental studies similar to this investigation include that of Patel, Goodling, and Khader [I] were experiments for monitoring the solidification front movement of water freezing within prisms of various geometries.

Grande [2] presented an analytical

solution for the time required to form an annulus of ice inside of a circular ice-water heat sink assuming the water is initially at the freezing temperature.

Lunardini [3] presented a solution for the time

required to freeze a liquid in a cylindrical container assuming zero initial superheat and compared his results with the results of others 21

22

W.E. Stewart, Jr. and K.L. Smith

for varying values of small Stefan number.

Vol. 14, Nc

Also in [4], Lunardini

presented the heat balance technique for predicting outward solidification around a cylinder.

Viskanta and Gau [13] studied the

inward solidification of three liquids in a horizontal cylinder over a wide range of Stefan numbers.

For small Stefan numbers, they found a

negligible influence of free convection upon the solidification rate. The heat balance integral method as described in [4,5,6] was used here as a basis to create an analytical model for the phase change problem, but with inward solidification in a cylinder.

Results have

been obtained from the analytical model for both the case of zero liquid superheat and the case of superheat in the liquid at the start of the process.

Superheat existed in the liquid (water) for all of the

experiments which were conducted.

Experiments In order to determine the solidification rate of the initially superheated liquid, data were collected from three experiments.

The

situation is shown schematically in Figure 1 and the experimental model is diagrammed in Figure 2.

The phase change liquid was distilled water.

The model was constructed from a 7.6 cm (3 in.) diameter copper tube which was three inches long with a 0.16 cm (0.063 in.) wall thickness. The copper tube was mounted in a clear Lexan housing which was used to contain the heat sink fluid (ethylene glycol-water) which was pumped across the outside of the tube. to the copper tube.

An air vent and fill tube were attached

A graduated scale was attached to the back of the

model to facilitate measurement of the phase change radius.

A flexible

tube was attached to the rear end plate along the cylinder axis to provide an escape for water which was discharged during the process as a result of thermal expansion. As the water froze from the inside surface of the copper tube toward the center, the radius of the phase change boundary was visually measured and recorded.

Photographs which depicted the phase change

radius in an axial view of the copper tube, were taken throughout the experiments at recorded time intervals.

The initial water uniform

superheat temperature was 22.2°C (70°F) for each of the experiments. Heat sink fluid temperatures of -17.8°C, -12.2°C, and

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~

SOLIDIFICATION C~ WATFR

23

-6.7°C (0°F, 10°F, and 20°F) were used.

Analytical Model To obtain the governing equations for calculating the radius of the phase change boundary as a function of time, the one-dimensional equation for radial conductive heat transfer was applied to the solid and liquid regions of the cylinder as shown in Figure i.

The cylinder

was assumed to be infinitely long and only conductive one-dimensional (radial) heat transfer was considered. of uniform liquid temperature (To).

The radius d encompasses an area

This temperature along with the

surface temperature of the cylinder (Ts), and the temperature at the phase change boundary (Tf), are assumed constant.

The radius (R) of the

freezing front extends from the center of the cylinder to the phase change boundary between the solid region and liquid region.

Both the

radius of the phase change boundary, R, and the uniform temperature radius, 6, of course decrease with time.

The surface temperature of the

cylinder is assumed to be lower than the freezing temperature and below the nucleation temperature. The quasi steady-state logarithmic approximation for temperature in the solid as suggested by Lunardini [4] was used: (T s - Tf) in (r/r o) T1 = T

s

-

(i)

in (R/ro)

with the following boundary conditions, T (ro,t) = Ts, T (R,t) = Tf, and

T (ro,O) = T O . The logarithmic approximation for the temperature of the liquid (T 2) as suggested by Lunardini [4], was also used: 6-r in (r/6) T 2 = r ° + (rf - To) (~/~)

(2)

with the boundary conditions T (R,t) = Tf, T (6,t) = T , for 6 > O, and o ~T 2-7 (6,t)

= 0.

The following equation is for the dimensionless phase change radius, 8, which is the same equation derived in [4] for outward solidification.

24

W.E. Stewart, Jr. and K.L. Smith

~

T

E

Vol. 14, N

CONSTANT

MPERATURE

" T " Te '

~ /

PHASE

CHANGE BOUNDRY

~T.T s

~" CYLINDER

WALL

FIG. 1 Schematic of Inward Solidification

FILL TUBE fAIR AIR VENT TUBE AIR VEN'r VALVE

I LIQUID EXPANSION TUBE

THERMOCOUPLE LOCATIONS' (8 PLS)

~COPPER

TUBE

--C COLD MEDIUM INLETS

COLD MEDIUM OUTLETS

--r-" COLD MEDIUM

~ C L E A R LEXAN HOUSING

FIG. 2 Experimental Set-up

Vol. 14, NO. i

INWARD SOLIDIFICATION OF WATER

82-I .... dS 1 48(-I-~ng)-Z'~P'~ ffi In8

[ST(2~n8

k21#(~l+l-~

I

25

(3) )

The energy equation for the liquid region, rearranged to a dimensionless form, resulted in,

(4)

dF ~2 ~2 d-~ = ~IST(R-I) + ~ISTInR

where: F

=~_+82

82 (l_~)in~ [ (I-~ z)

(5) + in ~ ( ~ - 5 )I +

~(a3_i)1

The resulting equations for F are the same as obtained by Lunardlni [4] for outward solidification. In order to solve for the dimensionless phase change radius, 8, and the uniform superheat temperature radius, R, as a function of dimensionless time, a fourth-order Runge-Kutta method [5] was used to numerically integrate Eq. (3) for dg/dT and Eq. (4) for dF/dT.

The

singularity for R was treated by letting R be 0.99 instead of 1.0 at time zero. Results The results from these three sets of calculations and experiments are also shown in Table I.

An uncertainty analysis was performed which

indicated the experimental values of the dimensionless solidification time, Tf, were within 13% of the actual values. Patel, Goodling, and Khader [I] conducted experiments, in part, for water freezing in a cylindrical container, except the cylinder was immersed in a cold bath.

The cylinder was 7.62cm (3 in.) in diameter

and constructed from 0.159cm (0.063 in.) wall copper. cylinder, however, was 2.54cm (I in.).

The length of the

In their experiments the initial

temperature of the water was at 0°C (32°F), the case of zero initial superheat.

A plot of 8 vs. T is shown in Figure 3 which compares the

experimental data obtained by Patel, et al., at T

= -16.5°C (2.3°F) and S

zero initial superheat with the experimental data we obtained for T

s

-17.8°C (0°F) and T

= 22.2°C (72°F). o

Figure 3 also contains the

26

W.E. Stewart, Jr. and K.L. Smith

Vol. 14

TABLE 1 Comparison of Experimental and Analytical Results

T S

= O°F i

T .02 .04 .05 .06 .07 .08 .]0 .12 .14 .15 .16 .17 .]8 .20 .22 .24 .26 .28 .30 .31 .32

T

= IO°F s = 0"0744 S~ 1.8180

= 0.10821. 2500 "" Ana3.- . . . . . . . . . 8 ~

Ex~ ~

.81 .74

.77 .73 .70 .67

.02 ,03 .04 .05 .06 .07 .08 i0 12 14 15 16 18 19 .20 .22 .24 .26 .28 .29 .30 .32

.68 .63 .58 .53 .47

.63 .57 .47

.42 .40 .37 .33 .28 .23 .18 .12 .06

.27

,14 o00

.00

T = 20°F Si = 0"0406 3.330

~

Anal. B

8

.81 .97 .80 .77

.74 .68

.67 .60 .53 ,47

.61 .55 .50 .45

.40 .39 .34 .33 .29 .24 .19 .13 .07

.27 .20

"'Exp.

Anal. B

.90 .83 .77 .73

.81

~ .02 .03 .04 .05 .06 .07 .08 .10 .12 .14 .16 .18 .20 .21 .22 .24 .26 .27 .28

.71 .64

.65 .58 .51 .47 .40 .35 .3]

.57 .52 .46 .41 .36 .30 .25

.28 .20 .13 .06

.20 .12 .00

.00

.13 .00 .00

TABLE 2 Results for Zero Initial Superheat

ST

Pesults of ~his Study

Lunardini[31

Rlleyf8]

Poots[9]

BeckettllO]

xf/£ T

~f/S T

Tf/S T

~f/S T

~f/S T

Allen & Severn~ll] Tf/S T

•050

5.60

5.00

5.25

3,47

5.30

....

.100

3,06

2.50

2.75

1.81

2.69

....

.200

1.81

1.25

1.50

.97

.250

1.56

I.O0

1.25

.81

. . . . . . . . 1.19

....

.641

.95

.39

.64

,40

....

.47

1.000

.80

.25

.50

.31

. . . . . . . .

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~SCK~IDIFICATIONOF~-~ZR

27

- - - E X P E R I M E N T A L DATA PATEL. GOODLING AND KHADER [I] (Ts=2,3 ° F , To-32"F, ST-O.IO04) EXPERIMENTAL DATA (Ts =0" F , To= 72 ° F, ST=0,1082 ) / 101.. 9.~ " ~ I~

--

ANALYTICAL CURVE (Ts=2.S'F , T.'32"F,

ST-O.IO04)

- - - - A N A L Y T I C A L CURVE (Ts=O'F, To =T2"F, ST=0.1082)

~.

! o'

0

.

.05

.

.tO

.

.

.15

.

.20

~c

.25

.30

,

.35

,

,40

,45

FIG. 3 Comparison of Experimental Results of [I] with Analytical and Experimental Results

"1'~

J I°i~

-RESULTS OF GRANDE'S EQUATION[2]

I\\

- A.,LYTICAL

CO.VE

.2'

a'.

'

'

'

%

FIG. 4 Comparison of Results of [2] with Present Results of ST = 0.1082

28

W.E. Stewart, Jr. and K.L. smith

results of the analytical model developed T

= -16.5°C

for zero initial superheat and

(2.3°F) and the analytical curve for T

equal to -17.8°C s The results from their experiments

s (O°F) and T

equal to 22.2°C (72°F). o and the experiments here do not compare favorably, part, to the differences Grande

Vol. 14,

in the experimental

due possibly,

in

techniques.

[2] presented an equation for the time required to form an

annulus of ice inside a circular ice-water heat sink, assuming the water is initially at the freezing temperature,

t

PiHRo 2 4K(TI_To )

as

1 .Rm) 2 in(~o)) ] [ -i~o (I-2

(6)

where t is time in hours, Ro is radius of cylinder in feet, Pi is density of ice, H is latent heat of fusion in BTU/Lb, T I is temperature at the phase change radius, T cylinder,

is temperature at the surface of the o K is thermal conductivity of the solid phase, Rm is phase

change radius Ro is radius of cylinder.

Equation

(6) was used to

calculate values of 8 vs. • for water with zero initial superheat and a cold medium temperature of -17.80C calculations

(0°F).

The results of these

are compared to the results of the analytical model for the

same conditions

in Figure 4.

In actual time, Grande's equation predicts

a 12 minute shorter freezing time than the heat balance integral method. Lunardini

[3] reported a value of ~f of 0.25 for the dimensionless

solidification container

time required for freezing water in a cylindrical

for zero initial superheat and contained a comparison of ~f/S T

vs. S T for the quasi-steady investigators.

approach and some results obtained by other

This comparison

is recreated

in Table 2.

The value of ~f is nearly a linear function of the Stefan number, approximately

related by,

Tf = 0.252 + 0.550S T

(7)

for a least squares curve fit. given by Riley,

et. al.,

This is somewhat similar to the relation

[8], as

• f = 2(1 + S T ) though the results here represent values of Tf having a greater dependency on Stefan number.

(8)

VOI. 14, NO. I

INWARD SOLIDIFICATION OF WATER

For the conditions used in the experiments, i.e. small Stefan numbers, the solidification process, was nearly radially uniform representing little influence by convective currents and was hence essentially "conduction-controlled" freezing [12]. Also, all of the cylinder boundary temperatures in the experiments were below the nucleation temperature of water.

The results of Viskanta and Gau [13]

also showed that convection was essentially negligible for S t < I, though the presence of convection was accounted for in their formulation of the heat balance equations. The analytical model does not consider the actual experimental conditions where, for practical purposes, the density difference between the solid and liquid phases forces the experimental procedure to allow for compression of the unfrozen liquid.

The liquid was allowed to

expand to atmospheric pressure through a vent, but the comparison to the analytical model is still very close.

Conclusions In general the analytical model presented appears to predict a longer dimensionless time required for various phase change radii than methods presented by past investigators.

However, the experimental

results for the dimensionless solidification time, ~f, were within 7 percent of the analytical results. For problems in which the temperature of the phase change fluid is initially at the fusion temperature, with zero initial superheat, Eq. (7) can be used to predict the solidification time as a function of the Stefan number.

At the relatively small Stefan numbers investigated,

free convection effects appear negligible.

NOMENCLATURE c k1

-

specific heat

- thermal conductivity of the solid thermal conductivity of the liquid

k2

-

k12 L

- kl/k 2 latent heat of fusion -

29

30

W.E. Stewart, Jr. and K.L. Smith

R

radius to phase change boundary

r

radius

Vol. 14, Nc

radius of the cylinder

r o

ST T

c (Tf - T )/L, Stefan Number

Tf T

freezing temperature of the liquid

S

temperature

constant liquid temperature at the

O

center of the cylinder temperature at the surface of the

T s

cylinder thermal diffusivity of the solid thermal diffusivity of the liquid

~2

R/r , dimensionless radius of phase O

change boundary radius of uniform superheat temperature region (~.t/r 2) ST , dimensionless time T ~- T ° o f , superheat parameter Tf - T s 6/R, dimensionless radius, superheat

T

region Subscripts 1

- solid region

2

- liquid region

REFERENCES 1.

Patel, G. S., Goodling, J. S., and Khader, M. S., "Experimental Results of Two-Dimensional Inward Solidification," Sixth International Heat Transfer Conference, Toronto, Canada (1978).

2.

Grande, E., Analysis-and Conceptual Design of Practical Ice-Water Heat Sinks, Corps of Engineers, U. S. Army, Cold Regions Research and Engineering LabOratory, Hanover, New Hampshire: Special Report 221, (1975).

3.

Lunardini, V. J., Heat Transfer in Cold Climates, Van Nostrand Reinhold Company, New York, N.Y., (1981).

4.

Lunardini, V. J., "Phase Change Around a Circular Cylinder," ASME Paper 80-WA/HT-5, Winter Annual Meeting, Chicago, (1980).

Vol. 14, No. 1

INWARD SOLIDIFICATION CFT~%TER

5.

Goodman, T. R., "The Heat-Balance Integral and It's Application to Problems Involving a Change of Phase," ASME Trans., 80, 335 (1958).

6.

Goodman, T. R., "Application of Integral Methods to Transient Nonlinear Heat Transfer," Adv. in Heat Transfer, Academic Press, I, 52 (1964).

7.

James, M. L., Smith, G. M., and Wolford, J. C., Applied Numerical Methods for Digital Computation, 2nd Ed., Harper and Row Publ., New York, N.Y. (1977).

8.

Riley, D. S., Smith, F. T., and Poots, G., "The Inward Solidification of Spheres and Circular Cylinders," Int. J. Heat Mass Transfer, 17, 1507 (1974).

9.

Poots, G., "An Approximate Treatment of a Heat Conduction Problem Involving a Two-Deminsional Solidification Front," Int. J. Heat Mass Transfer, 5, 339 (1962).

I0.

Beckett, P. M., Ph.D. Dissertation, Hull University, England, (1971).

II.

Allen, D. N. de G. and Severn, R. T., "The Application of Relaxation Methods to the Solution of Non-Elliptic Partial Differential Equations," Quart. J. Mech. Appl. Math., 15, 53 (1962).

12.

Sparrow, E. M., Ramsey, J. W., and Harris, J. S., "The Transition from Natural-Convection Controlled Freezing to Conduction-Controlled Freezing," ASME J. Heat Transfer, 103, 7 (1981).

13.

Viskanta, R., and Gau, C., "Inward Solidification of a Superheated Liquid in a Cooled Horizontal Tube," Warmeund Stoffubertragung, 17, 39 (1982).

31