An approximate method for calculating the freezing outside spheres and cylinders

J. P. GUPTA

Inthaunfrozen

zone (rf C r C R) T,=Tfatr=rf Tz=QatrBR.

The resulting temperature

(3)

protiles are:

Tf

‘0

T,-TT,_

l-To/r --roSrCrf 1 - h/r; Tw

Tf-

TW

(4) f

Tz _ 1 -Rlr F--l--~lr,’

Total heat evolved distance rf is

rfs

r
during freezing

4 Q =-~plL(rf3-rro3)+4~plc, _

(Tw--Tf)

x

3

rr”

(

rf2ro m2 2

2

up to the

Fig. 1. Freezing outside cylinder or sphere.

(rf3-ro3)

+

3

(9

>I

+$npzczTf(d+a-2)r~

(6)

where (Y= R/r, = constant. The heat evolved when the freezing front moves by a distance dr, is given by the differential of Eq. (6). This amount of heat which must be withdrawn at the cold surface in time dt is given by

mQf2drf+w~ Twrf2-(Tw- Tf) 3

[

(

3rf2 -rfro-2

ro2 >I

x drf + p2c2Tf ( a2 + (Y- 2) rf2drf =

kl

(T,- Tfhr (rf -ro)

fdt’

Fig. 2. Temperature profile during freezing outside cylinder or sphere.

(‘)

Integration of Eq. (7) from r. to rf and from 0 to t yields on rearrangement

The freezing front location can easily be calculated as a function of time since all the parameters in Eq. (8) are known. Values of the parameters used in this paper both for the spherical and the cylindrical freezing cases are listed in Table 1. Since no experimental data are available for freezing outside spheres, the value of (Y has been taken from Ref. [ 1 l] where it was derived from field experiments studying freezing outside a 78 ft deep cylindrical hole. The same value of (Ywas also found to hold good reasonably well for freezing outside a flat plate as determined by the author [ 133. Figure 3 shows the location of the freezing front as a function of time computed from Eq. (8). An-iterative analy-

A=($-$+~)[c,(~f--T,)+ (Tf2Tw) pIclro2 {1+=($+;-l)]] where p = rf/ro.

+i

(P2-P-1nP)

(8)

1630

Method for calculating the freezing outside spheres and cylinders FREEZING

Table 1. Parameters for figures (Y= 4-5 POCZ Set

ii

1 2 3 4 5 6 7 8 9 10

aY anY anY 1 1 1 2 2 2 any

Tf

L

CT,-TT,)

c,(T,- T,J

0 0 0 1 1 1 1 1 1 0

1 2 3 1 2 3 1 2 3 10

tical solution[ lo], available for set No. 10, is compared in the same figure (dotted line). The two match within 2 per cent though it is hard to say which of the two is closer to the true value. Heatflux density Heat flux density sphere is given by

OUTSIDE

CYLINDERS

Figures 1 and 2 also depict the cylindrical freezing case. The following equation from reference[ 111 for the cylindrical case corresponds to Eq. (8) for the spherical case

at the cold surface of the

Figure 4 shows the heat flux density as a function of time calculated from Eq. (9). For the parameters of Table 1, no data was available in the literature for comparison in this figure.

where P=

rflr0

Figure 5 shows the location of the freezing front as a funetion of time calculated from Eq. (10) for the parameter values listed in Table 1. Numerical values, available for the first six sets of parameters [4], are also compared in the same figure (dotted lines). For the fhst three sets, the deviation is within 4 per cent whereas for the remaining sets of parameters, the maximum deviation for short times is less than 15 per cent. At large times the comparison is much improved

Fig. 3. Frozen layer thickness outside sphere.. Tenth set shows comparison with iterative analytical solution (dotted line) of Shih and Chou [ 101to be within 2 per cent.

163 1

Method for calculating the freezing outside spheres and cylinders

I

2

456

3

7 89

0.75

0 25

Fig. 6. Heat flux density at the cold surface of the cylinder. First six sets show comparison with numerical solutions (dotted lines) of Tien and Churchill[4].

r,, radius of cylinder or sphere radius of freezing i radius of influence of the cold surface

good for large times, probably due to the neglecting of the unfrozen-phase conductivity. Even for these cases, the maximum deviation at short times is less than 15 per cent which is not too large when the uncertainties in the thermophysical properties of the wet soil systems can be as high as 20 per cent or even higher [ 131.

T T’ To T TlU

T’-T,,

real temperature initial temperature freezing temperature cold surface temperature

NOTATION A B

: L

; r

coefficient in Eq. (1) coefficient in Eq. (1) heat capacity thermal conductivity latent heat of freezing heat flux density at the cold surface total heat released radial distance

Greek symbols

cx Rlr, P rAr0 p density Subscripts

1 frozen zone 2 unfrozen zone

REFERENCES HI STEFAN J.,Ann. Phys. 189142 269. der mathemutischen Physik, 5th Edn., Dl RIEMANN G. H. and WEBER M., Die parriellen Di$erenrial-Gleichungen Vol. 2, p. 121. F. Vieweg and Sohn, Braunschweig 1912. [31 BANKOFF S. G., Advances in Chemical Engineering, (Eds. T. B. DREW et al.) Vol. 5, p. 75. Academic Press, New York 1964. 11790. [41 TIEN L. C. and CHURCHILL S. W.,A.Z.Ch.EJll965 PI TELLER A. S. and CHURCHILL S. W., Chem. Engng Prog. Symp. Ser. No. 59 1965 61185. J. E., Znt. J. Heat Mass Transfer 1970 13 123 1. Kl CHO S. H. and SUNDERLAND HASHEMI H. T. and SLIEPCEVICH C. M., Chem. Engng Prog. Symp. Ser. No. 79 1967 34 63. ;i; LAZARDIUS A., Int. J. Heat Muss Transfer 1970 13 1459. [91 SHIH Y. P. and TSAY S. Y., Chem. Engng Sci. 197126 809. HOI SHIH Y. P. and CHOU T. S, Chem. Engng Sci. 197126 1787. zumorazhivaniyu gruntov, Iadat el’stvo Akademii Nauk Hll KHAKIMOV K. R., Voprosy teorii i praktiki iskusstvennogo SSSR, Moskva 1957 (IT 66-5 105 1, U.S. Department of Commerce). [121 CARSLAW H. S. and JAEGER J. C., Conduction of Hear in Solids, 2nd Edn. Oxford University Press, London 1959. [I31 GUPTA J. P., Ph.D. Thesis, University of Pennsylvania, Philadelphia, Pennsylvania 1971.

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