- Email: [email protected]
Analytical solutions for freezing a saturated liquid inside or outside spheres
Chemical Engineering Science, 197 1, Vol. 26, pp. 1787-l 793.
Pergamon Press.
Printed in Great Britain
Analytical solutions for freezing a saturated liquid inside or outside spheres YEN-PING SHIH and TSE-CHUAN CHOU Chemical Engineering Department, Cheng Kung University, Tainan, Taiwan, China (Received 24 November 1970) Abstract-Using an analytical iteration technique successive solutions for the instantaneous frozen layer thickness and temperature profile for the freezing of a saturated liquid inside spherical containers with constant heat transfer coefficient are generated. Comparison of the results with numerical solutions shows satisfactory coincidence. Successive solutions for the freezing of a saturated liquid outside spheres are also derived. INTRODUCTION
with initial condition
IN A PREVIOUS
paper, Shih and Tsay [ I] applied the analytical iteration method proposed by Siegel and Savino[2-41 to study the solidification of a liquid at its freezing temperature inside or outside cylinders. The success of this method is due to a suitable transformation which transforms the diffusio’n equation in cylindrical coordinate into a convenient form. In this report, the same analytical iteration method is applied to study the transient solidification of a saturated liquid inside or outside spherical containers with constant heat transfer coefficient between the containers and the coolant. However, the success of this method is also due to the finding of a suitable transformation as given by Eq. (2). The results are then compared with the numerical solutions given in the literature [5]. FREEZING
OF LIQUID INSIDE CONTAINER
SPHERICAL
Consider a liquid at its freezing point inside a spherical container with coolant outside as shown in Fig. 1. The temperature of the coolant is maintained at To. Under the assumptions of constant physical properties of the solid and negligible volume change in solidification, the temperature of the solid T(r, t) and the instantaneous radial position of the solid-liquid interface R(t) can be described by the following system: g=&-~~(r$$),R(,
srcr,
R(f) = r.
,=_kc
(la)
dR k aT x=--g%,,
atr=R(t)
(lb)
T=
at r = R(r)
UC)
Tf
= h[T&) - T,,] ar Lo
at r = r,.
(14
Nomenclature is given at the end of the paper. Here the heat capacity of the spherical container has been neglected. The heat transfer coefficient h represents the sum of the resistance of the wall of the container and the film resistance between the container and the coolant. The temperature of the solid at the inner face of the container is Ti( t) , which is an unknown function of time t. The above system can be transformed into a convenient form for analytical iteration technique by defining the following dimensionless variables: m m #y- 1 -If
(1) 1787
26, No. I1 -A
t = 0,
and the boundary conditions
P
CES-Vol.
at
Tfu
To
i =Ti-T T,f= u(o9 r, Tf
7=
_
W’f- To) ro2pL
YEN-PING
SHIH and TSE-CHUAN
CHOU
[1], the iterative analytical solutions can be obtained easily. The details of the derivation are omitted. The iterative analytical solutions are: r=~[l+2~-(l+2~+3X)(l+X)-3]-~G(X) (4) (P+x)(l+X)-“-a% U(x,X)=-1+
dG’
(%
(p+X)(l+XPQ Here Fig. 1. Solidification of liquid inside spherical container.
Z(X,X)=BI,X(~+~)-~U(~,X)~~+~~~(~+~)-~
xv(t,X)d[+x
x=‘o-* r
J
,x(1 +5)-” W5,X)d5
(6)
and
x+1
G(X) = Z(x = X, X).
k
P=hr, Then Eqs. (I-ld)
(2)
become
dU cr(l+x)-4-=dr
lYW 8x2 ’
0 C x 6 x
(3a)
X(0) = 0 (*_~-4~=~~ x U(X, au axI,=,
(3)
7) = 0
(7)
The radial position of the solid-liquid interface X is given by Eq. (4) which is the inverse relationship of X(7). However, X has been assumed to be a monotonic function of r to assure the existence of the inverse relationship. Equations (4-7) provide an iteration procedure to obtain the instantaneous position of the solid-liquid interface and the temperature profiles. First, assume (Y= 0; this corresponds to neglecting of the time derivative term in Eq. (1). Then, Eqs. (4) and (5) become
(3b) ~,=~[l+2~-(1+2/3+3X)(l+X--3]
(8)
(3c)
(34
(9)
It is worth noting that the conventional transformation of the spherical coordinate system into the rectangular form by setting rT as a new variable has been found to be unsuitable for this analytical iteration technique. Using this conventional transformation the iterative analytical solutions are not convergent. By applying the procedure given previously
Here the subscript I denotes first-order approximation. Since the time derivative in Eq. (1) has been neglected, 71 and Ur are the quasisteady state solutions. Evaluating Ir(x, X) by substituting U,(x, x) into the integrands of Eq. (6). G,(X) = Zr(x = X, X), (aZ,/aX) and (dG,/dX) can also be obtained. The second-order approximate solutions can be obtained by the substitu-
= i [ I+
Z/$(r)].
1788
Analytical solutions for freezing a saturated liquid
tion of these formulae into Eqs. (4) and (5). That is rrr = rr - cG(X) (10) (p’+x)(l+x)-4-(Y~
Higher-order approximations can be obtained iteratively. The comparison of the analytical iteration solutions of the position of the solid-liquid interface with the numerical solutions obtained by Tao[5] is shown in Figs. 2 and 3. The results are quite coincident, except when values of both (Yand p are large. Since in practical condition, (Yand /3 are quite low, this exception is therefore not serious.
a11
U&X, x) = - 1+
(11) (~+x)(l+X+$$
where I,(~,~=p~~(l+5)-~(1*(~.~d~+~~g(l+6)-’
FREEZING
x~1(I,X)d~+x1:(1+5)~1,(5,X)d~
OF LIQUID
OUTSIDE
The solidification of a saturated liquid outside spherical containers as shown in Fig. 4 can be described by the following system:
(12)
G(X) = 11(x,x)1,=, =$+X)-‘[2+~-(1+2p)X
, ro c r c R(t)
-2(1 +x)-~l-~(l+x)-z. The third-order
approximate
SPHERES
(13) solution
(16)
with initial condition
~,&k’)
R(0) = r,
(16a)
is 7111 =
TI-
and the boundary conditions
(14)
aGd.4
where dR
G(X)
= I,x(P+5)(l+C)-4
u&,X)dt.
k aT
dt=pLar1,
(15)
?k(T,
-To)
I._ r.‘p
L
Fig. 2. Comparison of successive approximate solutions with numerical solutions of Tao[S]. I, II, III denote first-order, second-order and third-order approximations, respectively. Dotted lines are numerical solutions of Tao.
1789
(16b)
YEN-PING
SHIH
and TSE-CHUAN
CHOU
a
90.5
0.6
r.
tkff-1)
eP
L
Fig. 3. Comparison of successive approximate solutions with numerical solutions of Tao[5]. I, II, III denote first-order, second-order and third-order approximations, respectively. Dotted lines are numerical solutions of Tao.
technique. That is -(1-$4~=~;
0 Q x S x X(0) = 0 ,dx
41-m
x=7&
(17a) au
of liquid outside spherical container.
T(R, t) = T,
(17c)
1[l-t UdT)l axl,,o = p
(17d)
where the dimensionless as follows:
(164 ‘=
4
=-kg,
= h[T,-Tt(t)].
(1W
7%
The reasonable assumptions given for the derivation of Eqs. (l- 1d) have also been used. Equation (16d) has the same meaning as Eq. (Id). T,(t) is the unknown temperature of the solid at r,, the outside diameter of the spherical containers. The above system can be transformed into a suitable form for the analytical iteration 1790
(1W
U(X, 7) = 0
au
Fig. 4. Solidification
(17)
variables
T-TL T,-To
Ti-T Ui = Tf_ T,’ = u(“, d
r=-
kt
rK,p XC l_‘;p XC 1-g
are defined
Analytical solutions for freezing a saturated liquid
(dG/dX) in Eqs. (19) and (20). That is
L a = C,(T,- TO)
71=~[l-2p-(1-;y)-3+2(p+X)(1-_x)-31 p+-.
(18)
0
By applying the same procedure[l], iterative analytical solutions of the above system are presented as in the following:
(22)
U,(x,x) =
71+ G,(X)
(19) a(P+x)(l u&x, xl = - 1+
a(p+x)(l-X)-4+$$ C&,x)=-1+
-X)m4+$f (25)
-Xlm4+$$ where Z,(x, x) has been obtained by substituting U,(x, X) for V(x, X) into the integrands of Eq. (21). That is
where Zkx)
(24)
-Xlm4+$$
a(P+X)(l
(20) a(P+X)(l
(23)
These are the quasi-steady state solutions. The second-order approximate solutions are in=
+G(X)
;,“,.
= PI,xU -5)-“U(Z,x)dg+~~1(1-5)-”
xU(~,X)d~+xS:(1-5)-4U(5,X)d5 (21) z,(x,x)=$?+x)-'[-2-x+/3(1+2x) and
-(/3+x)(1
-x)-3+2(1
-x)-‘+(x-x)(1
g(A) = Z(x = x, x). The first-order approximate obtained by neglecting G(x),
solutions are (c~Z/&X) and
-x)-31 (26)
and G,(X) = ZI(x = X,x>.
I.6R r, I.4-
,k 0
Fig. 5. Frozen layer thickness in the solidification of liquid outside sphere.
1791
(27)
YEN-PING SHIH andTSE-CHUAN
CHOU
Fig. 6. Frozen layer thickness in the solidification of liquid outside sphere. The first three approximate solutions are almost identical.
The third-order approximate
a(P+x)(l U&,
-X)-4+%
x) = - 1 + @+X)(1
and spherical systems would give valuable results. Because of the simplicity and convergence of the analytical iteration technique, the development of this technique to more complicated systems would be of great-interest.
solutions are
-x)-4+
dC,, d_-$
(29)
Acknowledgements-The authors would like to thank Mr. S. Y. Tsay for discussion, the Computation Center of Cheng Kung University, and the National Science Council of the Chinese Government for financial aid.
where Z&x) can also be obtained by substituting U&,x) for z/,(x,X) into the integrands of Eq. (2 1). The instantaneous thicknesses of solid are computed from Eqs. (22) (24) and (28). The computational results are shown in Figs. 5 and 6. Notice that except for small a, the ratio of the latent heat to the heat capacity timed temperature difference, quasi-steady state approximations are quite accurate.
NOTATION
r
DISCUSSION
In this report and the previous one[l] the analytical iteration method of Savino and Siegel [2-41 is extended to systems with cylindrical and spherical configurations. Another analytical iteration method has also been proposed by Elmas [63 for a rectangular coordinate system. Comparison of different methods for cylindrical 1792
r0
T(x, t) Tf T,(t) To
parameter defined by Eq. (18) heat capacity of solid I(x = X, x) overall heat transfer coefficient defined by Eq. (6) or Eq. (2 1) thermal conductivity of solid latent heat heat flux across the container radial position of the solid-liquid interface radial distance inside diameter (Fig. 1) or outside diameter (Fig. 4) of the spherical containers temperature of solid freezing point temperature of solid at r = r. temperature or coolant
Analytical solutions for freezing a saturated liquid
U(x, T) dimensionless temperature of solid defined by Eq. (2) or Eq. (18) U*(T) dimensionless temperature at r = r, x dimensionless radial position defined by Eq. (2) or Eq. (18) X(T) dimensionless radial position of interface
Greek symbols
(Y,/3 parameters defined by Eq. (2) 4 dummy variable p density of solid 7 dimensionless time defined by Eq. (2) orEq. (18)
REFERENCES [l] SHIH Y. P. and TSAY S. Y., Chem. Engng. Sci. 197126 809. 121 SIEGEL R. and SAWN0 J. M., Proc. 3rd In?. Heat Transfer Conf., VoC 4, pp. 141-151. Am. Sot. Mech. Engrs., New York 1966. [3] SIEGEL R. and SAVINO J. M., Transient solidification of a flowing liquid on a cold plate including heat capacities of frozen layer and plate, NASA TN. D-4353 (1968). (41 SAVINO J. M., and SIEGEL R., Int. J. Heat Muss Transfer 1969 12 803. [5] TAO L. C.,A.I.Ch.E.JI 1967 13 165. [6] ELEMAS M., Int. J. Heat Mass Transfer 1970 13 1060.
Resume-Des solutions successives pour l’epaisseur de couches instantanement gelees et la courbe de temperature de congelation d’un liquide sature a l’interieur de recipients spheriques avec coefficient de transfert de chaleur constant, sont determinees au moyen dune technique d’iteration analytique. La comparaison des r&ultats avec des solutions numeriques montre une coincidence satisfaisante. Des solutions successives pour la congelation dun liquide sature a l’ext&ieur de spheres sont Cgalement derivees. Zusanunenfassung- Unter Verwendung einer analytischen Iterationsmethode werden fortlaufende Ldsungen fiir die Dicke einer momentan gefrorenen Schicht und das Temperaturprofil fiir das Frieren einer gesattigten Ldsung innerhalb von Kugelbefiltem mit konstantem W5rrneiibertragungsfaktor geliefert. Ein Vergleich der Ergebnisse mit numerischen Losungen zeigt zufriedenstellende Ubereinstimmung. Fortlaufende Liisungen fur das Frieren einer geslttigten Fliissigkeit ausserhalb von Kugelbehaltem werden ebenfalls erhalten.
1793
Pergamon Press.
Printed in Great Britain
Analytical solutions for freezing a saturated liquid inside or outside spheres YEN-PING SHIH and TSE-CHUAN CHOU Chemical Engineering Department, Cheng Kung University, Tainan, Taiwan, China (Received 24 November 1970) Abstract-Using an analytical iteration technique successive solutions for the instantaneous frozen layer thickness and temperature profile for the freezing of a saturated liquid inside spherical containers with constant heat transfer coefficient are generated. Comparison of the results with numerical solutions shows satisfactory coincidence. Successive solutions for the freezing of a saturated liquid outside spheres are also derived. INTRODUCTION
with initial condition
IN A PREVIOUS
paper, Shih and Tsay [ I] applied the analytical iteration method proposed by Siegel and Savino[2-41 to study the solidification of a liquid at its freezing temperature inside or outside cylinders. The success of this method is due to a suitable transformation which transforms the diffusio’n equation in cylindrical coordinate into a convenient form. In this report, the same analytical iteration method is applied to study the transient solidification of a saturated liquid inside or outside spherical containers with constant heat transfer coefficient between the containers and the coolant. However, the success of this method is also due to the finding of a suitable transformation as given by Eq. (2). The results are then compared with the numerical solutions given in the literature [5]. FREEZING
OF LIQUID INSIDE CONTAINER
SPHERICAL
Consider a liquid at its freezing point inside a spherical container with coolant outside as shown in Fig. 1. The temperature of the coolant is maintained at To. Under the assumptions of constant physical properties of the solid and negligible volume change in solidification, the temperature of the solid T(r, t) and the instantaneous radial position of the solid-liquid interface R(t) can be described by the following system: g=&-~~(r$$),R(,
srcr,
R(f) = r.
,=_kc
(la)
dR k aT x=--g%,,
atr=R(t)
(lb)
T=
at r = R(r)
UC)
Tf
= h[T&) - T,,] ar Lo
at r = r,.
(14
Nomenclature is given at the end of the paper. Here the heat capacity of the spherical container has been neglected. The heat transfer coefficient h represents the sum of the resistance of the wall of the container and the film resistance between the container and the coolant. The temperature of the solid at the inner face of the container is Ti( t) , which is an unknown function of time t. The above system can be transformed into a convenient form for analytical iteration technique by defining the following dimensionless variables: m m #y- 1 -If
(1) 1787
26, No. I1 -A
t = 0,
and the boundary conditions
P
CES-Vol.
at
Tfu
To
i =Ti-T T,f= u(o9 r, Tf
7=
_
W’f- To) ro2pL
YEN-PING
SHIH and TSE-CHUAN
CHOU
[1], the iterative analytical solutions can be obtained easily. The details of the derivation are omitted. The iterative analytical solutions are: r=~[l+2~-(l+2~+3X)(l+X)-3]-~G(X) (4) (P+x)(l+X)-“-a% U(x,X)=-1+
dG’
(%
(p+X)(l+XPQ Here Fig. 1. Solidification of liquid inside spherical container.
Z(X,X)=BI,X(~+~)-~U(~,X)~~+~~~(~+~)-~
xv(t,X)d[+x
x=‘o-* r
J
,x(1 +5)-” W5,X)d5
(6)
and
x+1
G(X) = Z(x = X, X).
k
P=hr, Then Eqs. (I-ld)
(2)
become
dU cr(l+x)-4-=dr
lYW 8x2 ’
0 C x 6 x
(3a)
X(0) = 0 (*_~-4~=~~ x U(X, au axI,=,
(3)
7) = 0
(7)
The radial position of the solid-liquid interface X is given by Eq. (4) which is the inverse relationship of X(7). However, X has been assumed to be a monotonic function of r to assure the existence of the inverse relationship. Equations (4-7) provide an iteration procedure to obtain the instantaneous position of the solid-liquid interface and the temperature profiles. First, assume (Y= 0; this corresponds to neglecting of the time derivative term in Eq. (1). Then, Eqs. (4) and (5) become
(3b) ~,=~[l+2~-(1+2/3+3X)(l+X--3]
(8)
(3c)
(34
(9)
It is worth noting that the conventional transformation of the spherical coordinate system into the rectangular form by setting rT as a new variable has been found to be unsuitable for this analytical iteration technique. Using this conventional transformation the iterative analytical solutions are not convergent. By applying the procedure given previously
Here the subscript I denotes first-order approximation. Since the time derivative in Eq. (1) has been neglected, 71 and Ur are the quasisteady state solutions. Evaluating Ir(x, X) by substituting U,(x, x) into the integrands of Eq. (6). G,(X) = Zr(x = X, X), (aZ,/aX) and (dG,/dX) can also be obtained. The second-order approximate solutions can be obtained by the substitu-
= i [ I+
Z/$(r)].
1788
Analytical solutions for freezing a saturated liquid
tion of these formulae into Eqs. (4) and (5). That is rrr = rr - cG(X) (10) (p’+x)(l+x)-4-(Y~
Higher-order approximations can be obtained iteratively. The comparison of the analytical iteration solutions of the position of the solid-liquid interface with the numerical solutions obtained by Tao[5] is shown in Figs. 2 and 3. The results are quite coincident, except when values of both (Yand p are large. Since in practical condition, (Yand /3 are quite low, this exception is therefore not serious.
a11
U&X, x) = - 1+
(11) (~+x)(l+X+$$
where I,(~,~=p~~(l+5)-~(1*(~.~d~+~~g(l+6)-’
FREEZING
x~1(I,X)d~+x1:(1+5)~1,(5,X)d~
OF LIQUID
OUTSIDE
The solidification of a saturated liquid outside spherical containers as shown in Fig. 4 can be described by the following system:
(12)
G(X) = 11(x,x)1,=, =$+X)-‘[2+~-(1+2p)X
, ro c r c R(t)
-2(1 +x)-~l-~(l+x)-z. The third-order
approximate
SPHERES
(13) solution
(16)
with initial condition
~,&k’)
R(0) = r,
(16a)
is 7111 =
TI-
and the boundary conditions
(14)
aGd.4
where dR
G(X)
= I,x(P+5)(l+C)-4
u&,X)dt.
k aT
dt=pLar1,
(15)
?k(T,
-To)
I._ r.‘p
L
Fig. 2. Comparison of successive approximate solutions with numerical solutions of Tao[S]. I, II, III denote first-order, second-order and third-order approximations, respectively. Dotted lines are numerical solutions of Tao.
1789
(16b)
YEN-PING
SHIH
and TSE-CHUAN
CHOU
a
90.5
0.6
r.
tkff-1)
eP
L
Fig. 3. Comparison of successive approximate solutions with numerical solutions of Tao[5]. I, II, III denote first-order, second-order and third-order approximations, respectively. Dotted lines are numerical solutions of Tao.
technique. That is -(1-$4~=~;
0 Q x S x X(0) = 0 ,dx
41-m
x=7&
(17a) au
of liquid outside spherical container.
T(R, t) = T,
(17c)
1[l-t UdT)l axl,,o = p
(17d)
where the dimensionless as follows:
(164 ‘=
4
=-kg,
= h[T,-Tt(t)].
(1W
7%
The reasonable assumptions given for the derivation of Eqs. (l- 1d) have also been used. Equation (16d) has the same meaning as Eq. (Id). T,(t) is the unknown temperature of the solid at r,, the outside diameter of the spherical containers. The above system can be transformed into a suitable form for the analytical iteration 1790
(1W
U(X, 7) = 0
au
Fig. 4. Solidification
(17)
variables
T-TL T,-To
Ti-T Ui = Tf_ T,’ = u(“, d
r=-
kt
rK,p XC l_‘;p XC 1-g
are defined
Analytical solutions for freezing a saturated liquid
(dG/dX) in Eqs. (19) and (20). That is
L a = C,(T,- TO)
71=~[l-2p-(1-;y)-3+2(p+X)(1-_x)-31 p+-.
(18)
0
By applying the same procedure[l], iterative analytical solutions of the above system are presented as in the following:
(22)
U,(x,x) =
71+ G,(X)
(19) a(P+x)(l u&x, xl = - 1+
a(p+x)(l-X)-4+$$ C&,x)=-1+
-X)m4+$f (25)
-Xlm4+$$ where Z,(x, x) has been obtained by substituting U,(x, X) for V(x, X) into the integrands of Eq. (21). That is
where Zkx)
(24)
-Xlm4+$$
a(P+X)(l
(20) a(P+X)(l
(23)
These are the quasi-steady state solutions. The second-order approximate solutions are in=
+G(X)
;,“,.
= PI,xU -5)-“U(Z,x)dg+~~1(1-5)-”
xU(~,X)d~+xS:(1-5)-4U(5,X)d5 (21) z,(x,x)=$?+x)-'[-2-x+/3(1+2x) and
-(/3+x)(1
-x)-3+2(1
-x)-‘+(x-x)(1
g(A) = Z(x = x, x). The first-order approximate obtained by neglecting G(x),
solutions are (c~Z/&X) and
-x)-31 (26)
and G,(X) = ZI(x = X,x>.
I.6R r, I.4-
,k 0
Fig. 5. Frozen layer thickness in the solidification of liquid outside sphere.
1791
(27)
YEN-PING SHIH andTSE-CHUAN
CHOU
Fig. 6. Frozen layer thickness in the solidification of liquid outside sphere. The first three approximate solutions are almost identical.
The third-order approximate
a(P+x)(l U&,
-X)-4+%
x) = - 1 + @+X)(1
and spherical systems would give valuable results. Because of the simplicity and convergence of the analytical iteration technique, the development of this technique to more complicated systems would be of great-interest.
solutions are
-x)-4+
dC,, d_-$
(29)
Acknowledgements-The authors would like to thank Mr. S. Y. Tsay for discussion, the Computation Center of Cheng Kung University, and the National Science Council of the Chinese Government for financial aid.
where Z&x) can also be obtained by substituting U&,x) for z/,(x,X) into the integrands of Eq. (2 1). The instantaneous thicknesses of solid are computed from Eqs. (22) (24) and (28). The computational results are shown in Figs. 5 and 6. Notice that except for small a, the ratio of the latent heat to the heat capacity timed temperature difference, quasi-steady state approximations are quite accurate.
NOTATION
r
DISCUSSION
In this report and the previous one[l] the analytical iteration method of Savino and Siegel [2-41 is extended to systems with cylindrical and spherical configurations. Another analytical iteration method has also been proposed by Elmas [63 for a rectangular coordinate system. Comparison of different methods for cylindrical 1792
r0
T(x, t) Tf T,(t) To
parameter defined by Eq. (18) heat capacity of solid I(x = X, x) overall heat transfer coefficient defined by Eq. (6) or Eq. (2 1) thermal conductivity of solid latent heat heat flux across the container radial position of the solid-liquid interface radial distance inside diameter (Fig. 1) or outside diameter (Fig. 4) of the spherical containers temperature of solid freezing point temperature of solid at r = r. temperature or coolant
Analytical solutions for freezing a saturated liquid
U(x, T) dimensionless temperature of solid defined by Eq. (2) or Eq. (18) U*(T) dimensionless temperature at r = r, x dimensionless radial position defined by Eq. (2) or Eq. (18) X(T) dimensionless radial position of interface
Greek symbols
(Y,/3 parameters defined by Eq. (2) 4 dummy variable p density of solid 7 dimensionless time defined by Eq. (2) orEq. (18)
REFERENCES [l] SHIH Y. P. and TSAY S. Y., Chem. Engng. Sci. 197126 809. 121 SIEGEL R. and SAWN0 J. M., Proc. 3rd In?. Heat Transfer Conf., VoC 4, pp. 141-151. Am. Sot. Mech. Engrs., New York 1966. [3] SIEGEL R. and SAVINO J. M., Transient solidification of a flowing liquid on a cold plate including heat capacities of frozen layer and plate, NASA TN. D-4353 (1968). (41 SAVINO J. M., and SIEGEL R., Int. J. Heat Muss Transfer 1969 12 803. [5] TAO L. C.,A.I.Ch.E.JI 1967 13 165. [6] ELEMAS M., Int. J. Heat Mass Transfer 1970 13 1060.
Resume-Des solutions successives pour l’epaisseur de couches instantanement gelees et la courbe de temperature de congelation d’un liquide sature a l’interieur de recipients spheriques avec coefficient de transfert de chaleur constant, sont determinees au moyen dune technique d’iteration analytique. La comparaison des r&ultats avec des solutions numeriques montre une coincidence satisfaisante. Des solutions successives pour la congelation dun liquide sature a l’ext&ieur de spheres sont Cgalement derivees. Zusanunenfassung- Unter Verwendung einer analytischen Iterationsmethode werden fortlaufende Ldsungen fiir die Dicke einer momentan gefrorenen Schicht und das Temperaturprofil fiir das Frieren einer gesattigten Ldsung innerhalb von Kugelbefiltem mit konstantem W5rrneiibertragungsfaktor geliefert. Ein Vergleich der Ergebnisse mit numerischen Losungen zeigt zufriedenstellende Ubereinstimmung. Fortlaufende Liisungen fur das Frieren einer geslttigten Fliissigkeit ausserhalb von Kugelbehaltem werden ebenfalls erhalten.
1793