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Experimental measurements and a numerical method for ice

sublimation

Aguirre-Puente, J.; Sakly, M.; Goodrich, L. E.; Lambrinos, G.

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Ser

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National Research

Conseil national

LO.

1335

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2

B

m

Division of

Division des

--

Building Research

recherche~

en betiment

Experlmental Measurements

and a Numerical Method for

Ice Sublimation

by J. Aguirre-Puente,

M.

Sakly, L,E. Goodrich

and

G.

Lambrinos

ANALYZED

Reprinted from

Proceedings of the Fourth International Symposium on

Ground Freezing

Sapporo, Japan. August 5

-

7,1985

Volume 2, p. 1 - 8

(DBR Paper No. 1335)

Price $2.50

NRCC 25197

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La sublimation de la glace dans les milieux poreux pose des

problemes dans un certain nombre de domaines de 11ing6nierie.

Ce document expose les fondenents thgoriques ngcessaires pour

dgcrire mathQmatiquement les phenodnes en jeu, fournit des

donnges sur les vitesses de sublimation de la glace pure et

dQcrit brisvement un modele numgrique de calcul des vitesses de

sublimation dans des conditions non isothermes.

(4)

Fourth International Symposium on Ground Freezing/Sapporo/S-

7

August 1985

Experimental measurements and

a

numerical

method for ice sublimation

J. AGUIRRE-PUENTE & M. SAKLY

Laboratoire d'A~rothemzique du C. N. R. S, 92190 Meudon, France L. E. GOODRICH

National Research Council, Div. of Building Research, Ottawa K I A OR6, Canada

G. Lambrinos

National Agronomic Institute of Athens, 11855 Athens, Gr2ce

Abstract

Problems involving sublimation of ice in porous media arise in a number of engi- neering contexts. This paper reviews the theoretical basis necessary for a mathema- tical description of the phenomena concer- ned and presents data on sublimation rates for pure ice. A numerical model for rompu- ting sublimation rates in non-isothermal conditions is described briefly.

Introduction

In response to concerns of the Food Engineering Industry. on one hand. and Ener- gy storage and Artic Engineering interest. on the other hand, during the last five years the "Laboratoire d1Aerothermique du C.N.R.SW has studied sublimation prwblems in ice and mfrozen dispersed media.

In fact, sublimation studies carried out to date have been largely concerned with biological porous medLa in the

context of freeze drying. Consequently, re- search has, for the most part, concentra- ted on sublimation at low pressures 11,2

1

But in order to reduce the high cost asso- ciated with such drying procedures new techniques are constantly being sought, and a potentially interesting avenue is offered by freeze drying at atmospheric pressures.

Moreover, sublimation of free ice or ice contained in porous media arises in

certaln Civil Engineering problems. In parti- cular, a problem of present day interest is that of liquid gas storage at atmospheric pres- sure in underground cavities where the ther- mal and mechanical behaviour of the frozen cavity walls, constituted by an ice layer on frozen soil or rock, requires the study of the sublimation process acting upon the sur- faces exposed above the liquid gas in the re- servoir during very long periods 13

1

.

Sublimation problems also arise in the context of the dehydration of stored frozen food-stuff s 14

1

of medical substances, the thermal balance and thermal regime in perma- frost and at natural ice surfaces, the thermo- mechanical behaviour of structures in arctic conditions, the behaviour of interstellar ice, etc.

A considerable amount of experimental data has already been obtained and we now can provide information concerning the funda- mental knowledge of ice as well as certain aspects useful for pratical applications.

Starting from general physical and mathe- matical models of the sublimation process,

we suggest in this paper simplifications and adaptations required to interpret the expe- rimental data and we indicate the most urgent. problems remaining to be investigated. A nu- merical model for general sublimation pro- blems is presented and, finally, a new expe- rimental device conceived to study ice-subli- mation phenomenon in the absence of convec- tion and under very low temperatures and

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relative humidities is described briefly. Experimental study

In order to study the sublimation pro- blem in the laboratory, an insulated tunnel was constructed where temperature T, velocity V and relative humidity RH of the cooled air flowing in a closed circuit could be regulated.

The tunnel is composed of four linear sections placed in a vertical plane. The 20cm x 20cm test section is located in the upper horizontal portion and is preceded and followed by a convergent and a diver- gent system respectively, used to ensure suitable air flow conditions around a verti- cal cylindrical sample undergoing sublima- tion. The samples of approximate dimensions 7cm high by 2,6cm diameter are placed between two insulated metallic supports, one of which is removable to permit the sample to be weighed periodically in order to follow the dehydration process.

Details of the experimental device are given in previous papers )5,6

1

.

The range of conditions which can be explored is as follows : 0,5 < V < 4 m s-'

,

243 K < T < 264K, 30 < RH < 90%.

Ice, vegetal and animal tissues (pota- toes, meat) and a biological simulation sub- stance (methyl-cellulose) have been studied so far (6,71. Experimental data is recorded and reduced to various types of diagrams the most useful of which give the sublima- tion flux vs.velocity, relative humidity and temperature. An example is shown in fi- gure 1 which corresponds to a set of tests made on pure ice.

A discussion of various fundamental and applied aspects of the previous research is contained in several published papers

18.9

(

.

The information obtained so far, for pure ice makes it possible to establish an empirical relation ship between the sublima- tion flux and the air velocity and tempera- ture. Calculations were made for several values of relative humidity but here we men- tion only the results for 70% RH.

To illustrate the type of relation ship obtained using the well known method of se- paration of variables, the data were fitted to curves of the form

\?here A = exp - (24.2932 + 0.0989 T)

and B = 1.013~.(-1.2689 + 5.4138.10- T) This equation correlates quite well with the experimental data. Most of the results lie within a few percent of the correspon- ding curves (Fig. 1). The values for 258K

diverge, however, by as much as 18% and ap- pear to be anomalous 1101

.

Theoretical

The mechanism governing the sublimation of ice in frozen dispersed media have been fully discussed elsewhere )7,8

1

and will be only briefly described here. Physically, the sublimation process arises when molecules within a solid arrive at the surface with a thermal energy sufficient to allow their departure to the surrounding medium, aba- lance being established between molecules departing and those returning. Sublimation isthan on-going process which can occur at any temperature below the melting point of the solid. The vapour produced is removed by vapour transfer processes such as dif- fusion and convection while the rate of vapour production is conditionned by the thermal conditions imposed. The process is thus one of simultaneous heat and mass trans- fer in the presence of a non constant space and time dependent vapour source which consu- mes part or all of the solid ice phase. Mass transfer rates are determined by moistu- re and temperature gradients as well as by the thermophysical properties of the disper- sed medium.

While, in frozen soils or biological materials, the fact that the ice present is dispersed throughout the material matrix mo- difies somewhat the final result, neverthe- less the case of sublimation from a pure ice surface is fundamental to understanding the more complex problem and its study may be justified on these grounds alone, aside from its interest for the applications already noted.

Mathematical statement

A

three-zone general model for sublima- tion within frozen dispersed media has been proposed earlier 18,9

1

.

This model is gover- ned by simultaneous heat and mass transfer with a moving transition zone separating the dehydrated from the non affected zone. The equations have poorly known non-linear thermophysical coefficients and source and sink terms which make the model difficult to apply. In this paper we study the case where the sublimation transition zone can be reduced to a moving front and we consider the extreme case of dispersed media, namely that of pure ice. Assuming the diffusion of vapour and heat are governed respectively by Fick and Fourier's laws, the set of equa- tions can be written for the ambient medium :

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Figure 1 : Ice

T E M P E R A T U R E ( K )

(7)

and for ice :

awl 0 . - = o

1 at

,

(no vapor transfer), (5)

where pa. and Aa, Aiare the density and

thermal conductivity of air and ice respec-

tively, D the coefficient of -pour diffu-

sion in ay?, ca and c. the heat capacity of air and ice respec$ively, w the mass

per unit volume of moist air, T the

temperature, x the space coordinate and t

the time.

The model does not formally take into account forced convection at the interface. This can be done, however, using the charac- teristics of the diffusion boundary layer corresponding to the velocity considered.

At the moving interface, x = S(t),

vapor is produced at a rate given by

wkere p. and wF are the mass per unit volu-

me of ie: and water vapour at the interface.

In generel, the sublimation flux term contains contributions arising from simple diffusion in air Id, from convective pro-

cesses I

,

as well as a term If to account

for possyble ablimation, and we have

Straight forward mass and heat balance

considerations at the interface yield :

and

In addition, vapor saturation is assumed to prevail at the interface and a relation between saturation vapor content and tem- perature is assumed known in order to com- pletely specify the problem. Diffusion is taken to be through a stagnant air film constituting the diffusion boundary layer. The mixture of dry air and water v q m r is

assumed to obey the ideal gas law. Numerical method

Equations (2) to (5) can be discretized using either finite element or finite dif- ference procedures. Taking linear basis functions with lumped capacitance matrix terms in the finite element formulation or considering the heat balance over a fini- te volume surrounding a node in the finite difference formulation, leads to the follo- wing set of nodal equations for the regions

above and below the moving interface :

for nodes i 4 iu-1 and

for nodes i 5 iu-1 and i 3 id+l, where iu

and id are the nodes delimiting the element m, containing the moving sublimation front,

~"nd Tn are the nodal values of moisture

coBcentration and temperature respectively

at time-level n, and the remaining material

andmesh parameters are defined in figure

L .

Ineqns (9a) (9b) material properties may be evaluated at time-level n to avoid nodal iterations, this being adequate in most cases. Time-stepping is controlled by

the parameter 8 where :

1 , < e < 1

9

The moving interface is treated by a local moving-mesh method in which the element containing the moving interface is tempora- rily subdivided into two sub-elements with distinct material properties. Geometrical non-linearly is confined to the moving in- terface element only and the original mesh

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definition is retained throughout the cal- culation. P

smf-l

= C. (x. -x. ) lU 1u-1 Kmf-l = At element m < mf

*

Dmf-l = At.D/(xiu-~iu-l)

Xiu, Hiu, Tiu iu

Xf, Hf, Tf ... element m Smfd = Cd. (x. ~d lu -X. ) f

Kmfd = At .kd/ (x. -x. )

id iu id Xid. Hid, Tid

'mf+l = C'(Xid+l-xid)

element m > m Kmf +l = At .k/(xid+l-xid) f'

Figure 2 : Mfinition of mesh parameters

Similarly to equations (9a) (9b), heat where backward time differences have been and mass balance equations at nodes iu and used for simplicity.

id can be written : At the moving interface, equations (8)

and (9) correspond to : n+ 1 i[sel+g

.smfui

.(T:;~-T:~) = I n L and (xid-x ) .Wo.L. (gn+l-gn) = ( T ; * ~ - T ~ ~ ~ ) / ~ ~ ' ~ iu

I

K~~-~.(T:~~~-T::~)+K~~~.

(T;:~-T?'

)

Ty

1 -Tn+ 1 (lob) ld f m u gn+l +Kmfd' 1 (lib) n+ 1 n+l n and

1

[Smf+,+il-g ).Smfdl .(T. -T. ) = 7 ld ld , L

Equations (10) and (11) form a simple n+l n+l n+ 1 n+l) type of local moving mesh method in which '~d(~f -Tid )fKmf +l.(~~:l-~id the mesh defomtion is confined to a single

element. A solution technique originally

(lot) devised for the heat solution equation with

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suitable modifications, be extended to the Table 1 was calculated for the following present case. This technique involves the

simultaneous use of both forward and back- Values Ax = 0'01 * = l-'

'

=

ward Gaussian elimination formulations for the exception of a small initial error tridiagonal matrices, given a relation the values are nearly indistinguishable for betweenwf and T

,

equs (10) and (11) can pratical purposes.

then be expressei entirely in terms of

known coefficients. The primary non- Discussion

linearityof the problem 'is thus isolated w

and, for constant or sufficiently slowly In order to analyse sublimation pro- varying material properties, an iterative blems in general we require information about solution technique is required only for the the diffusion conditions at the surface, Interface element equations. Ha,ving corn- and the experiments carried out to date can puted gn+l, T ~ l w;;l, , T;;l and T%l the provide some insight. Using the measured ave-

rage sublimation flux and supposing a linear remaining nodal temperatures and moisture distribution of vapour concentration near the contents can be calculated directly by back surface the diffusive boundary layer thick- substitution using the appropriate Gaussian ness

Gdiff.

can be estimated from Ficks law. elimination coefficients. As a result, the From hydrodynamic theory

1

13

1

we can method is very efficient and, because it is also calculate a thickness for the dynamic a moving mesh formulation, the accuracy is boundary layer,& From the experimental excellent and is relatively insensitive to data, for velocit$gg'ranging from 1.5 to the size of mesh spacing used, in contrast 4 m.s-

,

the ratio 6 /6 was found to with the more usual methods. vary linearly bet~eebq.3~Afifi 4.4 for tem-

A complete description of this method temperature -30°C and between 2.3 and 2.7 is given in a separate paper

1

12

1

which for temperature -9°C. Until more thorough also includes a number of general results studies can be carried out which take account obtained with the model. As an indication of local sublimation rates and the real geo- of the numerical accuracy of the technique, metry of the specimens, we suggest that the- Table 1 compares the surface position as se ratios can be used to estimate the thick- a function of time computed by the model, ness of the diffusion boundary layer from with the analytic solution for a simple the dynamic boundary layer thickness for o- Neumann problem. For an initial concentra- ther sublimation problems in the range of tion p., the surface x = 0 being held at temperatures and velocities studied. This concentration

w

ambient, the progression should make it possible to exploit more ful- of the sublimation front is given by the ly the numerical model presented for more dimensionless equation : general cases of porous media and non-iso-

thermal conditions.

L Conclusion

where h is the root of

In this article we have presented an ex- \j;;.G.~.e~~-erf (1)

+

1 = 0 perimental study of the sublimation of pure

Dva.t. ice and have proposed a numerical model to

and F is the Fourier number, Fo =

-.

solve the system of equations which describe

L~ sublimation phenomena in a more general con-

text. It is however necessary to verify the L is a scale length and G = (pi- wf)/ model by comparison with non-isothermal ex-

(W ambient-

w

)

f periments which are more elaborate than those TABLE 1 : Comparison of numerical and analytical solutions for sublimation depth.

1 1

x

error error 1 I 1 x lo-2 5 x

lo-2

Fo (Fourier No)

6

0.114 x 10-I

-

-

-0.383 x 10-1 0.124 0.278 S/L calculated 0.124 x 10-1 - 8.5 0.392 x 10-1 -9.37 - 2.4 0.124 +3.54

+

0.0 0.277

1

+8.39 x + 0.3

1

S/L analytic

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carried out to date. At the same time it is also necessary to study the range of condi- tions most important for modern engineering concerns, namely very low temperatures with relative humidities near zero. To this end we have recently designed a new experimen- tal system. This consists of an experimen- tal cell in which the diffusion of the su- blimated vapour takes place in a nearly dry nitrogen atmosphere and under a one-dimen- sional temperature gradient directed upwards to avoid natural convection.

In addition it is necessary to have at hand a more thorough knowledge of the dif- fusive boundary layer. This will require further experimental work with the wind tun- nel, measuring local sublimation rates on cylindrical specimens suitably arranged, for example, to allow control of the zone expo- sed to sublimation.

Along with this experimental work, a theoretical study of thermal, diffusive and dynamic boundary layers around a cylinder is also envisaged.

References

1 D. Simatos, G. Blond et P. Dauvois 1974 La lyophilisation, A.N.R.T, Paris 2 A.V. Lykov, and D.P. Lebedev 1973 Study

of the ice sublimation process. Int. J. of Heat and Mass Transfer, 16, 1087-1096.

3 A. Boulanger et W. Luyten 1983 Le stocka- ge souterrrain a basse temperature et ses applications. C.R. XVIo Cong. Int. du Froid Paris ,1, 365-370.

4 T.J.R. Cooper 1970 Control of weigh los- ses during drilling, freezing. storage and transport of pig meat. Bull. de IIF/ IIR, Annexe 1970-3, 175-181, 187-190. 5 G. Lambrinos, J. Aguirre-Puente et M. Giat

1983 Une soufflerie pour l'etude experi- mentale des milieux disperses soumis a des temperatures negatives. Int. J. of Refrig., 6 , 1.

6 G. Lambrinos 1982 Deshydratation des mi- lieux disperses pendant leur congelation et leur conservation a l'etat gele. The- se Ing- Doct, Paris VI.

7 R.N. Sukhwal 1984 Physique de la sublima- tion en milieu disperse et experimenta- tion sur la viande et la tylose. These Ing-Doct, Paris VI.

8 J. Aguirre-Puente and R.N. Sukhwal 1984 Sublimation of ice in frozen dispersed media. 3rd Int. Symp. on Offshore Mech. and Artic Engng, New Orleans, U.S.A. 9 J. Aguirre-Puente, G. Lambrinos and M.

Sakly 1984 Sublimation of ice in frozen dispersed media : Physics phenomena, equations and experimental study. Proc. of the Coll. Int. "Probl6mes a frontieres libres" Maubuisson-Carcans

,

France to be pub. by Pitman.

10 J. Aguirre-Puente, G. Lambrinos and

M.

Sakly 1984 Experimental research of the sublimation rate of ice samples. 10th An- nual Meeting of European Geophys. Soc. to be pub. in Ann. Geophysicae. 11 L.E. Goodrich 1978 Efficient numerical

technique for one-dimensional thermal pro- blems with phase change. Int. J. of Heat and Mass Transfer, 21, 615-621.

12 L.E. Goodrich et al. Anumerical model for heat and mass diffusion with a moving su- blimation front.(in preparation).

13 H. Schlichting 1955 Boundary layer theory Pergamon Pr London.

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T h i s paper. v h i l e being d i s t r i b u t e d i n r e p r i n t form by t h e D i v i s i o n of B u i l d i n g Research, remains t h e c o p y r i g h t of t h e o r i g f n a l p u b l i s h e r . It s h o u l d n o t be r e p r o d u c e d i n whole o r i n p a r t w i t h o u t t h e p e r m i s s i o n of t h e p u b l i s h e r . A l i s t of a l l p u b l i c a t i o n s a v a i l a b l e from t h e D i v i s i o n may be o b t a i n e d by w r i t i n g t o t h e P u b l i c a t i o n s S e c t i o n . D i v i s i o n of B u i l d i n g R e s e a r c h , N a t i o n a l R e s e a r c h C o u n c i l of C a n a d a . O t t a w a . O n t a r i o . K1A 086. Ce document e s t d i s t r i b u O s o u s forme d e t i r e - & p a r t par l a D i v i s i o n d e s r e c h e r c h e s e n batisrent. Les d r o i t s d e r e p r o d u c t i o n s o n t t o u t e f o i s l a p r o p r i € t € d e l ' s d i t e u r o r i g i n a l . Ce document n e p e u t B t r e r e p r o d u i t en t o t a l i t s ou en p a r t i e s a n s l e consentement d e l l € d i t e u r . Une l i s t e d e s p u b l i c a t i o n s d e l a D i v i s i o n p e u t e t r e o b t e n u e en O c r i v a n t

a

l a S e c t i o n d e s p u b l i c a t i o n s , D i v i s i o n d e s r e c h e r c h e s e n b a t i m e n t . C o n s e i l n a t i o n a l d e r e c h e r c h e s Canada, O t t a v a , O n t a r i o . K1A 0R6.

Figure

Figure  1  :  Ice
Figure  2  :  Mfinition of mesh parameters
TABLE 1  :  Comparison of numerical and analytical solutions for sublimation depth.

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