M A Boles

M N Ozisjk

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Exact Solution for Freezing in Cylindrically Symmetric, 
Porous, Moist Media 
M. A. Boles1 and M. N. Ozisjk2 
Nomenclature 
Kl2 = thermal conductivity ratio, — 
k2 
L = dimensionless latent heat, 
' c2T0 
Q = dimensionless strength of the line 
sink, 
k,Tn 
S(7) 
R = dimensionless radial coordinate, — 
the dimensionless location of 
s(t) 
freeze-front,-
' 0 
U(R,T) = nond imens iona l mois tu re 
concentration for the region 
r > S ( t ) ,
U ^ ^ 
a = diffusivity 
ay = diffusivity, «; 
A = nondimensional thermal gradient 
coefficient, —— 
dj(R,7) = nondimensional temperature for 
the regions ; = 1 and ; = 2, ——— 
' o 
6U = nondimensional freeze-front 
temperature, —^-
7 = nondimensional time, 
r0
2 
Subscripts 
l,2,m = refers to solid, liquid, and 
moisture, respectively. 
Introduction 
Experimental investigations [1, 2] of freezing in a porous 
medium, such as a wet soil, have shown that the freezing is 
accompanied by moisture migration towards the freezing 
front. The migration of moisture effects the temperature 
distribution in the medium and the location of the freezing 
front. Freezing in a wet porous medium has numerous 
engineering applications. For example, during the extraction 
Mechanical and Aerospace Engineering Department, North Carolina State 
University, Raleigh, N. C. 27650, Mem. ASME 
Mechanical and Aerospace Engineering Department, North Carolina State 
University, Mem. ASME 
Contributed by the Heat Transfer Division for publication in the JOURNAL OF 
HEAT TRANSFER. Manuscript received by the Heat Transfer Division May 13, 
1982. 
of energy from the earth using a heat-pump, earth-coupled 
thermal system, the cooling pipe buried in the earth can be 
regarded as a line heat sink in an infinite medium. As the 
energy is extracted from the moist porous soil, the moisture is 
redistributed, lowered in temperature, and becomes frozen. It 
is of interest to know the temperature distribution and the 
location of the freezing front. Also, in the measurements of 
thermal conductivity in moist porous media, the knowledge of 
the moisture distribution upon freezing is important. 
The standard formulation of phase-change problem 
without allowing for the movement of moisture within the 
medium is not applicable for the analysis of this type of 
freezing problems. More complicated phase-change problems 
allowing for the movement of moisture have been studied in 
connection with evaporation [3] and freezing [4] only for a 
semi-infinite medium. The studies of freezing due to a line 
heat sink in an infinite medium with cylindrical symmetry are 
also limited [5, 6] and do not allow for moisture movement. 
In the present study exact solutions are developed for the 
temperature and moisture distribution and the location of the 
freezing front caused by cooling with a line heat sink in an 
infinite medium with cylindrical symmetry. 
Formulation of the Problem 
Consider a line heat sink of strength, q, (W/m) positioned 
at the origin of the radial coordinate, r = 0, in a moist, 
porous, infinite medium. Initially the medium is at a uniform 
temperature, T0, and contains moisture of uniform 
distribution, u0. At time t = 0, the heat sink is activated. The 
freezing process begins and the freezing front propagates in 
the radial direction. Let r — s(t) be the position of the freeze 
front at any time, t. The dimensionless temperature 
distribution 0,(.R,T) in the freezing zone, 0 < R < S(7), where 
there is no moisture is governed by the equation 
30, (R,7) 1 3 
: « 1 2 ( * ^ - ) , i n O < J ? < S ( 7 ) , r > 0 (1) 
dr " " R dR 
In the unfrozen region S(7) < R < oo, the dimensionless 
temperature distribution 62(R,7) and the moisture 
distribution U(R,T) are governed by the following system 
obtained as a special case from the Luikov's system of 
equations (87) and (88) of [7] 
dd2(R,7) _ 1 3 
~ R 3R 9T 
dU(R,7) _ 1 d 
~~~~17~ ~a'"2R dR 
(R~),mS(7)<R<o°,7>0 
( dU \ 1 9 / d62 \ 
{R~dW)-^AR3RKR^R)' 
(2) 
mS(7)<R<°°,7>0 (3) 
It is to be noted that in the Luikov system [7] the zones 1 and 2 
refer, respectively, to evaporation and moisture zones, 
whereas in the present system they refer to solid and liquid 
zones, respectively. The boundary and initial conditions 
become 
dd{ 
~3R. 
el(s,7) = e2(s,7)--
£im (2TR-±)=Q 
R-O \ o  / 
AV 
36 1 (S,T) 
dR 
dU(S, 
dR 
362(S,7) 
3R 
,7)_Ad67 
62(R,0)' 
U(R,0) = 
e2(oc,T)--
U(oo,T)--
= L[\-
,(S,7) 
dR 
= 1 
= 0 
= 1 
= 0 
-U(S, 
= 0 
vds 
T)]d7 
(4) 
(5) 
(6) 
(7) 
(8a) 
(8b) 
(9a) 
(9b) 
Journal of Heat Transfer MAY .1983, Vol. 105/401 Copyright © 1983 by ASME
  
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1.0 
0.5 
I 0. 
© 
-0.5 -
-1.0 
-1.5 
1 1 l i p 
_ev=o.5 
L = 1.2 
A • 1.0 
°„z*0-4 
Kl2=1.0 
a„= 1.0 
\ H ' 
^-W-W^ 
U) 
I 
ScUJI ' 
e ^ \ ^ 
J& 
111', i 
I l i | I l 
i4 
V 6.0 
111' I I 
l l | I 
Q 
,-2.0 
^4.0 
<6.0 
i iNi 
1 1 , J 
111 
-
\ -
111 
10 10 10" 
a , t 
10 10 10 
Fig. 1 Effect of nondimensional line heat strength, 0, on temperature, 
0, and moisture, U 
I 
-0.5 
© 
1 J I M i l I M i l I M i l I M i l I I 
10 10 10 10 10 10 
a , t 
Fig. 2 Effect of nondimensional freeze-front temperature, 0V, on 
temperature, 0, and moisture, U 
3 
l 
© 
0.5 
0. 
0.5 
1 0 
1.5 
1 1 1 Ip^K 
Q =4.0 
ev =0.5 
A =1.0 
ami = 0 . 4 
Kl2 1.0 
- a,2 = 1.0 
— 
(-I-U) 
I [ I I I j 
, 2 k 6 I , i 
J l 11 
6 
_ ^ 
-U+L 
1 1 M | 1 
^ 0 . 4 
i nil i 
M i| i 
L 
2.0 
1.2 
0.4 
-U-rM-
I I I 
-
' 
_ 
Y\V 
-444-
10 10 10 10 10 
a o t 
Fig. 3 Effect of nondimensional latent heat, L, on temperature, 
moisture, U 
1.0 
10 
, and 
0.5 
3 
I 
-0 .5 -
-1 .0 
-1 .5 
10 
1 
-4 
1 1 1 |~-K.. 
Q »4.0 
«v "0.5 
L - 1 . 2 
a„,«0.4 
K „ * 1 . 0 
« „ - 1 . 0 
(- I-U) 
I 111 i I 
III 
9^~ 
^ 
-r+L 
I I I 
^ 
-M-r^ 
I I 
A 
- 1 . 5 
- 1 . 0 
- 0 . 5 
- 0 . 0 
-M-
"1 
A 
- 0 . 0 
- 1.5 
HU-
I I! 
V -
m 
10 10 10 10 10 
• t » 
Fig. 4 Effect of nondimensional thermal gradient coefficient, A, on 
temperature, e, and moisture, U 
Solution of the Problem diffusion type homogeneous equations, (1), (2), and (11), 
Equations (2) and (3) can be decoupled by defining a new subject to the boundary and initial conditions given above, 
variable Z(R,T) as [8] These solutions are determined as 
Z(R,r) = 62(R,r) + CU(R,r) (10a) e{R>r) = e + Q\Ei(__*_) 
where 
C= 1 - a-ml (106) 
After some manipulation the equation for Z(R,T) is deter-
mined as 
dZ(R,T) 1 9 / BZ 
5 = a'"2 
OT 
^ ( * ^ - ) , i n S ( T ) < * < o o > T > ( ) (11) M * . D = l + 0 . - l ) 
-£ , / ( -X 2 ) l , inO<«<S(T) 
'(-4) 
(12) 
Ei(-\2ai2) 
,inR>S(T) (13) 
Now the problem has been reduced to the solution of three and 
402/Vol. 105, MAY 1983 Transactions of the ASME Downloaded From: https://heattransfer.asmedigitalcollection.asme.org on 06/29/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
U(R,T) = 
where 
Vm2& 
/ ( - X 2 a 1 2 ) La,„; 
am2 , 2 
l - a m 2 Ei( 
X=-
S(r) 
(14) 
(15) 
2(al2T)'
A 
Finally, the transcendental equation for the determination of 
X is obtained by introducing the solutions given by equations 
(12), (13), (14) together with the definition (15) into the in-
terface energy balance equation (6). We find 
Q ,2 e„-i 
47T Ei(-\2al2) 
••\2a„L 1 
1 ~ aml ? 2 
A r 
— ( » B - i ) 
• ot,„2 L 
e "ml 
£ / ( - X 2 a 1 2 ) 
£7(-X2tv„„)-a:,„2Jj 
(16) 
Results and Discussion 
Once the constant X is computed from the solution of the 
transcendental equation (16) for a given set of system 
parameters, the location of the freeze-front S(T) at any time r 
is determined according to equation (15), the temperature 
distribution 0l (R,r) in the freeze-zone is calculated from 
equation (12), and the temperature distribution 82{R,T) and 
the moisture distribution U(R,T) outside the freeze-zone are 
calculated from equations (13) and (14), respectively. 
Sample calculations are performed to illustrate the effects 
of various system parameters on the temperature and 
moisture distribution in the region. These results are 
presented in Figs. 1-4 by plotting the temperature and the 
moisture content as a function of the dimensionless parameter 
T/R1 = a2t/r
2. The physical significance of such figures can 
be interpreted in two different ways: 
1 For a fixed value of time, the curves represent the 
variation of temperature or moisture as a function of the 
radial position, r, in the medium. Then, on these figures the 
origin r = 0 corresponds to a location a2t/r
2 — oo, and the 
right-hand side of the curves correspond to the frozen zone. 
The curves for the moisture content show that the moisture is 
highest at the freeze-front; moving away from the freeze-front 
with increasing r, there exists a region where moisture content 
falls below the initial value, followed by the zone where the 
initial value of the moisture content remains unchanged. 
2 For a fixed value of the radial position, r, the curves 
represent the variation of temperature or moisture with time. 
Then, a2t/r
2 — 0 corresponds to the beginning of the cooling 
process. As time increases, the temperature at a given location 
continuously decreases; on the other hand, the moisture 
content falls below the initial value for a short period, then 
begins to increase continuously until the freezing front 
reaches to that location. 
Having discussed the general behavior of the variation of 
temperature and moisture as a function of time and position 
in the region, we now focus our attention to the effects of 
various system parameters. 
Figure 1 illustrates the effects of the nondimensional line 
heat strength, Q, on temperature and moisture. As the sink 
strength is increased, the freeze-front moves deeper into the 
medium at a given time; or the freeze-front arrives much 
quicker at a given location. Increasing Q, lowers the values of 
temperature and moisture. 
Figure 2 illustrates the effects of the nondimensional freeze-
front temperature, 6V, on the temperature and moisture 
distribution in the medium. Decreasing dv, increases the 
maximum moisture content and moves the freeze-front 
towards the origin, r = 0. 
Figure 3 illustrates the effects of nondimensional latent 
heat, L, on the temperature and moisture distribution. 
Lowering L forces the freeze-front further away from the 
origin, r = 0. The minimum and maximum moisture contents 
do not seem to depend on L. 
Figure 4 shows the effects of the nondimensional thermal 
gradient coefficient, A, on the temperature and moisture 
distribution. The maximum moisture content is strongly 
dependent upon A; increasing A increases the maximum 
moisture content. The temperature level and the freeze-front 
location, however, are very little affected by A. 
Acknowledgment 
This work was supported in part by the National Science 
Foundation through the Grant MEA-8110 705. 
References 
1 Aguirre-Puente, J., and Philippe, A., "Quelques recherches effectuees en 
France sur le probleme de la congelation des sols," Rev. Gen. Thermique, Vol. 
96, 1969, pp.1123-1141. 
2 Aguirre-Puente, J., Dupas, A., and Philippe, A., "Quelques resultats 
concernant le gonflement au gel et leur application au classement des sols selon 
leurdegre'degelivite',"Bu//. liasonLabo. P.etCh., Vol.68,1973, pp. 23-29. 
3 Mikhailov, M. D., "Exact Solution of Temperature and Moisture 
Distributions in a Porous Half-Space With Moving Evaporation Front," In-
ternationalJournal of Heat and Mass Transfer, Vol. 18, 1975, pp. 797-804. 
4 Mikhailov, M. D., "Exact Solution for Freezing of Humid Porous Half-
Space," International Journal of Heat and Mass Transfer, Vol. 19, 1976, pp. 
651-655. 
5 Patterson, S., "Propagation of a Boundary of Fusion," Proceedings of 
the Glasgow Mathematics Association, Vol. 1, 1952, pp. 42-47. 
6 Ozisik, M. N., and Uzzel, J. C , "Exact Solution for Freezing in Cylin-
drical Symmetry With Extended Freezing Temperature," JOURNAL OF HEAT 
TRANSFER, Vol. 101, 1979, pp. 331-334. 
7 Luikov, A. V., "Systems of Differential Equations of Heat and Mass 
Transfer in Capillary-Porous Bodies (Review)," International Journal of Heat 
and Mass Transfer, Vol. 18, 1975, pp. 1-14. 
8 Mikhailov, M. D., "General Solution of the Diffusion Equations Coupled 
at the Boundary," International Journal of Heat and Mass Transfer, Vol. 16, 
1973, pp. 2155-2164. 
9 Lebeolev, N. N., Special Functions and Their Applications, Prentice-Hall, 
Inc., Englewood Cliffs, N.J., 1965. 
Natural Convection From Needles With Variable Wall 
Heat Flux 
J. L. S. Chen1 
Nomenclature 
a = dimensionless needle size 
b = needle shape parameter, 2(m -
l)/5 
/ = dimensionless stream function 
g = gravitational acceleration 
Gr = modified Grashof number, 
g(3q0L
4/(kp2) 
h = convection heat transfer co-
efficient 
Associate Professor, Department of Mechanical Engineering, University of 
Pittsburgh, Pittsburgh, Pa. 15261, Mem. ASME 
Contributed by the Heat Transfer Division for publication in the JOURNAL OF 
HEAT TRANSFER. Manuscript received by the Heat Transfer Division July 23, 
1982. 
Journal of Heat Transfer MAY.1983, Vol. 105/403 Downloaded From: https://heattransfer.asmedigitalcollection.asme.org on 06/29/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use


Exact Solution for Freezing in Cylindrically Symmetric, Porous, Moist Media

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Exact Solution for Freezing in Cylindrically Symmetric, Porous, Moist Media

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