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A model of the thermal conductivity of porous water ice at low gas pressures
Pinner. Printed
Space Sci., Vol. 39, No. 3, pp. 507-513, in Great Britain.
A MODEL POROUS
00324633/91 03.00f0.00 Pergamon Press plc
1991
OF THE THERMAL CONDUCTIVITY OF WATER ICE AT LOW GAS PRESSURES GERHARD
Institut
fur Weltraumforschung,
STEINER and NORBERT I. K6MLE
iisterreichische Akademie der Wissenschaften, A-8042 Graz. Austria
Observatorium
Lustbiihel,
(Received in$nal form 5 September 1990) Abstract-This study is concerned with energy transfer in porous water ice at large Knudsen numbers of the sublimated gas. We present a formula for the effective thermal conductivity, &s, which is found to be valid for a wide range of material parameters, in particular also for high porosity materials, where other expressions for &published in the literature lose their validity. It is found that above m 190 K &increases strongly with temperature and depends mainly on the average grain size, while at low temperatures it is primarily controlled by the value of the Hertz-factor. The influence of the porosity is relatively weak. The reliability of our expression for I,, has been checked by comparing computed temperature profiles with the temperature recordings of comet simulation experiments. The predicted temperatures are in good agreement with the experimental results. Finally, our model is applied to a typical surface layer of a cometary nucleus. A strong temperature dependence of the effective thermal conductivity in cometary water ice is predicted.
1. INTRODUCTION
Water
ice is present
exists
below
in comet
the surface
nuclei
of Mars
and probably and
minor
In all studies where the ice is assumed to be porous it is pointed out that the vapour phase significantly contributes to the energy transfer. Here we discuss the problem of energy transfer in porous water ice within the scope of laboratory experiments. In Section 2 we present a new approach to calculate the effective thermal conductivity of porous water ice at low gas pressures which is based on the work of Zehner and Schliinder (1970). In order to demonstrate the validity of the model it is applied to comet simulation (KOSI) experiments (Section 3). In Section 4 the dependence of Izelfon the material parameters is discussed, and the relevance of our results for the thermal properties of cometary surface layers is noted briefly.
also planets
(Fanale et al., 1986, 1989). For comet Halley H,O was found to comprise more than 80% of the volatile outflow, and there is evidence that water is the dominant volatile in most observed comets (Brownlee, 1989). Although coma observations do not directly relate to the composition of the nucleus, sublimation of water ice and heat transfer into its interior is thought to be the driving mechanism for the evolution of cometary nuclei. Hence, insight into the physical processes related to the transfer of energy in water ice is a necessary precondition for reliable model calculations. This fact is reflected in the new generation of comet models which largely depend on the physical properties of water ice. In a number of studies related to the evolution of comets the thermal properties of compact water ice are used to compute temperature profiles in the cometary interior &linger, 198 1; Kiihrt, 1984; Herman and Podolak, 1985). Smoluchowski (1982) first took into account that comet nuclei are probably highly porous bodies and derived an expression for the effective thermal conductivity of a porous icy nucleus utilizing the classical Maxwell equation. Squyres et al. (1985) computed temperatures within comet nuclei using an equation first given by Brailsford and Major (1964) for the effective thermal conductivity of the material. A different approach was taken by Espinasse (1989) and Spohn and Benkhoff (1990), who considered a model of coupled heat and mass transfer within an ice matrix.
2. HEAT CONDUCTION
MODEL
The one-dimensional, time-dependent heat conduction equation in a porous body with temperaturedependent material properties is given by
Here $ is the porosity of the material, pice is the mass density of compact ice, ci, is the heat capacity of compact ice, 1,s is the effective thermal conductivity, T is temperature, x is the depth below the ice surface, and t is time. Equation (1) can be solved for T with appropriate boundary and initial conditions, provided that a suitable expression for I,, is available. In the following we will derive an expression for I,e at low gas pressures by treating porous water ice without 507
508
G. STEINER and
N. I. KOMLE
admixture of dust or other volatiles as a two-phase material, where the two phases are the ice grains and the void spaces. 2.1. Thermal conductivities of the single phases In the SI system the thermal conductivity of crystalline water ice is given by (Klinger, 1981) li,
= 567/T.
‘E
(2)
Energy transport in the voids occurs via thermal radiation and the sublimated gas. At large Knudsen numof a void space, bers lvold, the thermal conductivity can be derived as follows : consider two planes of ice separated by a distance S and kept at temperatures T and T+AT, respectively. The energy removed from a plane at temperature T per unit area and unit time is given by
FIG. 1. THERMAL AND
CONDUCTMTY OF COMPACT ICE (CURVE 1) OF A VOID SPACE FOR ,'$= lo-' Ill(CURVE 2) AS A FUNCTIONOFTEMPERATURE.
Curves 3 and 4 are the contributions of thermal radiation and the transfer of latent heat to AVoid,respectively.
l/Z
P(2kT+L’).
(3)
Here E is the infrared emissivity of the ice, Q is the Stefan-Boltzmann constant, k is the Boltzmann constant, m is the mass of a gas molecule, L’ is the latent heat of sublimation per molecule, and P is the vapour pressure given by the ClausiusClapeyron equation P = a embiT,
(4)
where a and b are two experimental parameters (see Fanale et al., 1986). In writing down equation (3) it was assumed that the sublimation of H,O ice into vacuum can be described by the Hertz-Knudsen formula. This assumption has been experimentally verified by Tschudin (1946). The first term in equation (3) is the emitted thermal radiation, the second term comprises the energy carried by the gas molecules, where 2kT denotes their mean thermal energy (Knudsen, 1946). For AT << T*/b, the energy flux between the planes can be linearized around T. Consideration of the leading contributions and comparison with the Fourier heat conduction equation gives AvoIdas
“* dP dTLS,
(5)
where E’ is the infrared emissivity corrected for multiple reflections and L denotes the latent heat of sublimation per unit mass. Note that the contribution of the transfer of internal energy to the energy flux, both by conduction and advection, is negligible compared with the latent heat effect. For other geometries of the void different geometric factors have to be inserted into equation (5) (see Steiner, 1991). However, their influence on Avoidis not significant and will not be
further discussed here since, as will be seen, we will express I,, as a parametrized equation. In order to illustrate the relative importance of li, and AvoidFig. 1 shows the thermal conductivities of the single phases as a function of temperature for S = lo- 3 m. 2.2. Effective thermal conductivity Many different relations have been suggested in the literature to determine the effective thermal conductivity of a two-phase material. A classical formula is due to Maxwell, who derived 1,, for a dilute suspension of spheres of conductivity 1, embedded in a continuum of conductivity 12*by solving the Laplace equation for temperature around one sphere. The resulting expression for leff is first order accurate in tj, and hence only correct in the limit $ -+ 0. Notwithstanding this restriction many workers have used Maxwell’s result as though it were valid for all porosities. However, at high porosities it is not clear which phase is to be taken as continuous, and l,e depends strongly on this choice. This is demonstrated in Fig. 2, which shows leff as a function of temperature for $ = 0.5 and S = lo-’ m. In curve 1 the ice phase was chosen as continuous, and in curve 2 the void spaces were taken as the continuous phase. It can be seen from the figure that in the investigated temperature range & closely follows the thermal conductivity of the continuous phase. This observation provides a natural explanation for the circumstance that the effective thermal conductivity of porous water ice as given by Smoluchowski (1982), who utilized Maxwell’s equation with the ice phase as the continuous medium, is not suitable to explain the temperature curves recorded in comet simulation experiments in the big space simulator (see Spohn and Benkhoff, 1990). Note that a similar argument as to
509
Thermal conductivity of porous water ice at low gas pressures
water ice at large Knudsen numbers of the sublimated vapour can be written as &I = (1 -X/=&%~d+%,/=?
B is the “deformation
factor” which controls particle shape and is related to porosity via .
T
(ii1
FIG. 2. EFFECTIWTHERMALCONDUCTIVITYOFPOROUSWATER ICE AS A FUNCTION OF TEMPERATURE CALCULATED FROM MAXWELL'SEQUATION. In curve I the ice is considered as the continuous phase ; in
curve 2 the void spaces are considered
as the ~ntinuous phase. Also shown are ,I,, (curve 3) and ,I,,, (curve 4).
Maxwell’s equation applies to Russel’s equation, which yields similar results as Maxwell’s equation, and was used by Espinasse (1989). An updated version of Maxwell% approach has been obtained by Jeffrey (1973) who derived an expression for &.s which is second order accurate in $. However, this method also only applies to small porosities. For high porosity materials the effective thermal conductivity can be derived by simulating the thermal conductivity by a combination of parallel and series connections of “thermal resistances”. The simplest approach is to either average conductivities or resistivities or use a weighted sum of parallel and series connections (see Parrot and Stuckes, 1975; Tsotsas and Martin, 1987). An elaborate model to calculate the effective thermal conductivity of a system of particles in contact surrounded by a fluid, which has been tested against a wide set of data, was worked out by Zehner and Schltinder (Tsotsas and Martin, 1987 ; Zehner and Schliinder, 1970, 1972). They consider a unit cell characteristic for the material, which consists of an outer fraction att~buted to the fluid phase only and a core covered by both the solid and the fluid phase, and derive &t by an electric circuit analogy as indicated above. Their model accounts for heat conduction in the fluid phase, the solid phase, heat transfer over particle contact points and thermal radiation. An expression for a, in porous water ice at low gas pressures can be obtained from their model [see Tsotsas and Martin, 1987 ; equation (S)] by substituting lLra,,,the radiative thermal conductivity of a void space, by lvold and taking the limit L + 0, where 1 denotes the thermal conductivity of the fluid phase. Thus, the effective thermal conductivity of porous
(h is the so-called flattening coefficient. It determines the amount of heat flowing through particle contact points and is equivalent to a Hertz-factor which accounts for the reduced heat &IX in granular material compared with compact material. Note that in the derivation of equation (6), S, the average dimension of a pore space, has been replaced by $d, where d denotes the average particle diameter (Zehner and Schlunder, 1972). Thus, d is the relevant length scale which has to be used in dete~ining ;iyoid in the expression for &. Equation (6) consists of three terms that contribute to the effective thermal conductivity of the material: the first term accounts for energy transfer in the void spaces, the second term for heat transfer over particle contacts, and the third term for the energy transferred in the core of the unit cell which is covered by both the fluid and the solid phase. A wide range of flexibility to determine & for different materials is provided by three adjustable parameters that characterize the investigated substance. These parameters are the flattening coefficient 4, the average particle size d, and the porosity 3/. 3. APPLICATION
TO COMET
SIMULATION
EXPERIMENTS
Recently the thermal evolution of ice samples under space conditions has been studied by laboratory experiments in two space simulators at the DLR/Koln (F.R.G.) (for a review see Griin et al., 1989). In the following we report about a series of experiments in the small vacuum chamber carried out within the framework of the comet simulation (KOSI) programme. The main purpose of these experiments was to study the influence of a “porous crust” on the thermal evolution of the ice sample. 3.1. Experimental description The sample container was a 10 cm sized box with Teflon walls and a copper back plate. The samples consisted of frozen droplets of water ice, were covered
G.
510
and N. I. K~MLE
STEINER
Y
T [I<) FIG. FREE
3. KNUDSEN PATH
TEMPERATURE
OF
NUMBER THE
FOR
GAS
s =
S/S, WHERE MOLECULES,
lo-’ m
6 DENOTES AS
CALCULATED
A
THE MEAN
FUNCTION FROM
OF
KINETIC
GAS THEORY.
The vapour pressure was calculated from equation (4). The region between the dashed lines is the transition region between the free molecular and continuum flow regime as given by Zucrow and Hoffman (1976).
FIG.
4a.
COMPUTED
OVER A TIME PERIOD
CIllBELOW WITH
THE
TEMPERATURE OF
4h
ICE SURFACE
THE MEASURED
3.2. Model calculations Now we report the computations for one experiment which is taken as representative for the whole series. Equation (1) has been solved for temperature with Aerrexpressed by equation (6). The applicability of equation (6) is demonstrated in Fig. 3, where the Knudsen number of sublimated water vapour is shown as a function of temperature for S = lo- 3 m. Note that the plotted curve provides a lower limit for the Knudsen numbers occurring in the experiments. The temperature of the ice surface (upper boundary) was approximated by
TO
OF 1,
(SOLID
LINES)
2, 3, 5
AND
BOTTOM)
cuRvEs
COMET SIMULATION
T, = Ti+(T,‘-ps) by steel plates containing an array of equidistant holes simulating a porous crust, and were irradiated by an artificial sun. The chamber pressure was kept at x lo- * Pa. The back plate of the sample container was cooled to - 120 K. The ice temperature was recorded over 5-6 h at a depth of 1,2, 3, 5, and 7 cm below the ice surface. The main effect of the crust was to raise the surface temperature, depending on the porosity of the crust, in comparison with a freely sublimating ice surface. At the uppermost temperature sensor a constant temperature was reached after -30 min. After 4 h the samples were nearly isothermal down to more than 75% of their depth. The temperature curves within the samples attained the convex shape that has also been observed during experiments in the big space simulator (Spohn and Benkhoff, 1990). The porosity of the sample material was determined as $ x 0.5, the average grain size was estimated to lie in the range of 0.1-I mm. The experiments are described in more detail in KGmle et al. (1991).
(TOP
TEMPERATURE
FOR A REPRESENTATIVE
EVOLUTION
AT A DEPTH
tanh(t/r),
7
COMPARED
(DASHED
LINES)
EXPERIMENT.
(8)
where TS is the initial surface temperature which is obtained by linear extrapolation from the initial temperature below 1 cm depth, T,’ is the temperature at the uppermost temperature sensor after a steady state has been reached, and z is a time constant typical for the warming up period of the surface. At the lower boundary the measured temperature of the back plate has been used. The initial temperature profile has been produced by linear interpolation between the temperatures determined at the positions of the temperature sensors. Equation (1) has been solved for temperature by an implicit numerical code. The result was checked by recalculating the solutions with an explicit finite difference method. The porosity was taken as $ = 0.5, which is in agreement with the values determined experimentally. The adjustable parameters were chosenas~=4~10~~,d=O.l6mmandr= 16min. This choice gave the best fit to the recorded temperatures. It should be noted that for the other experiments the adjustable parameters vary at most by a factor of three. Figure 4a shows the computed temperature evolution (solid lines) over a time period of 4 h at a depth of 1, 2, 3, 5 and 7 cm below the ice surface in comparison with the measured temperature profiles (dashed lines). The accuracy of the computed temperatures is within 10% of the observed temperatures. A systematic deviation from the recorded temperatures occurs at the two lowermost temperature sensors. This feature is observed in all experiments and is speculatively attributed to radial heat fluxes which are not considered in the model. Figure
Thermal
conductivity
of porous
water ice at low gas pressures 100
FIG. 4b. COMPUTED SPATIAL TEMPERATURE DISTRIBUTION IN THE ICE SAMPLE CORRESPONDING TO FIG. 4a AFTER 0.5,1, 1.5, 2,3 AND 4 hINTOTHEEXPERImNT (LERTORIGHT). The measured temperatures are indicated by dots.
FIG. 6. INFLUENCE OF THE AVERAGE PARTICLE SIZEON &AS PREDICTED BY EQUATION (6). 4 = lo-*, ((I =0.5 and 10m6 m < d-c 10m2 m (bottom to top). d is increased by a factor of 10 between two neigh-
bouring
curves.
resulting curves for the effective thermal are in good agreement in the investigated range and for the utilized parameters. 4. DISCUSSION
T
IKI
FIG. ~.EFFECTI~ETHERMALCONDUCTIVITYOFTHEICESAMPLE PERTAINING TO FIG. 4a AND b COMPUTED FROM EQUATION (6) (SOLID LINE).
Also shown are the curves for & calculated from the equation of Linsky (dashed dotted line) and Brailsford and Major (dashed line).
4b shows the spatial temperature profiles for the same experiment at z = 0.5, 1, 1.5, 2, 3 and 4 h computed from equation (1) (solid lines) compared with the measured temperatures (dots). Figure 5 shows the thermal conductivity of the ice sample pertaining to Figs 4a and b computed from equation (6). It is seen that at low temperatures the derived leff- closely follows c#&. At temperatures higher than approximately 190 K it strongly increases with temperature and eventually exceeds lice. Also shown in Fig. 5 are the curves computed after Linsky (1966) whose method to calculate 1, (adopted to the current problem) consists of simply adding &, and &,id, and the equation of Bra&ford and Major (1964), again with the thermal conductivity of the solid phase taken as $J&. As can be seen from the figure the
conductivity temperature
AND CONCLUSIONS
We have derived an expression for the effective thermal conductivity of porous water ice at low gas pressures and utilized it to reproduce the temperatures recorded in comet simulation experiments. Now it is interesting to discuss in more detail the influence of the adjustable parameters, 4, d and r,G, which characterize the investigated material, on the effective thermal conductivity according to equation (6). The effect of the average particle size d is shown in Fig. 6, where &is displayed for 10m6m < d < lo-* m and fixed values of 4 and +. It is seen that the average particle size significantly affects 1,, at temperatures higher than approximately 170-210 K. At low temperatures 1,, is mainly controlled by the value of the flattening coefficient since the conductivity of the voids is small at low temperatures. This is displayed in Fig. 7, where &is plotted for fixed d and $ and lo- 5 < r$ < 1. From Fig. 8, where lefl is shown for fixed d and 4 and two extreme values for the porosity, it is obvious that the effect of porosity on the effective thermal conductivity is small compared with the influence of d and 4. Note that while the influence of the average particle size and the flattening coefficient on l,= as predicted by equation (6) is in good agreement with the corresponding predictions from the equations of Linsky and Brailsford and Major, different conclusions are drawn concerning the influence of porosity. Linsky’s expression on the one hand is independent of $, the
G. STEINER and
N. I. K~~MLE
000,
’
1111111(111
T CKI FIG. ~.INFLUBNCEOFTHEFLATTENING COEFFIC~ENTON,&AS PREDICTED BY EQUATION (6).
d= 10e3 m, + = 0.5and lo-’ < C#J< 1 (bottom to top). C/J is increased by a factor of 10 between two neighbouring CWWS.
FIG.
8.
INFLUENCE OF POROSITY ON lcffAS PREDICTED BY EQUATION (6). d = lo-' m, 4 = lo-‘, $I = 0.9 (solid line) and + = 0.1
(dashed line). equation of Brailsford and Major on the other hand exhibits a similar behaviour to Maxwell’s equation at high and low porosities. Thus, it predicts that at high volume fractions of the void spaces 1,, tends to zero with Avoidat low temperatures (see Fig. 9). This behaviour is in contradiction to the observation that individual grains may form high porosity agglomerates of particles in contact over whose contact areas heat flow can occur so that 1,s is expected to be proportional to 4iice at low temperatures. This line of evidence supports the validity of equation (6), and we will use it to discuss briefly a possible scenario for the thermal conductivity of water ice present in comets. Greenberg has proposed the possibility that mostly sub-micrometre-sized dust grains covered by refractory organics and volatiles constitute the building elements of comets. The size of these composite particles is suggested to lie in the micrometre range (see
FIG. 9. EFFECTIVETHHWALCONDUCT~VITYASAFUNCTIONOF TEMPERATURE COMPUTED FROM EQUATION (6) (SOLID LINE), THE EQUATION OF LINSKY (DASHED DOTTED LINE) AND THE EQUATION OF BRAIL~FORD AND MAJOR (DASHED LINE) FOR d = 10m3 m, cj~= lo-’ AND + = 0.9.
FIG. 10. POSSIBLE THERMAL CONDUCTIVITY OF THE NEAR SURFACEREGIONOFACOMETARYNUCLEUS(SOLID LINE).
Also shown is the thermal conductivity (dashed line).
of compact ice
Greenberg, 1986). The value of the Hertz-factor can be expected to be much smaller in the cometary situation (probably below 10e5) than was found in our experiments, which were performed in a one-genvironment. Figure 10 displays the thermal conductivity of porous water ice as it may be typical for the near surface layer of a cometary nucleus, which is already depleted of the more volatile species. The effect of a probable admixture of dust particles is not considered here. The parameters chosen are d = 10m6 m, $ = IO-’ and $ = 0.8. Also shown in Fig. 10 is the thermal conductivity of compact water ice in the same temperature range. It is clear from the figure that l,e of such a material is completely different from li,. At low temperatures it is proportional to 4&c ; at temperatures higher than approx. 190 K it increases with temperature and varies by about three orders of magnitude in the relevant temperature range.
Thermal
conductivity
of porous
Acknowledgements-This work has been supported by the Austrian “Fonds zur Fijrderung der wissenschaftlichen Forschung” under grant number P7246-GEO. Furthermore we wish to thank the members of the KOSI team for providing the opportunity to perform the experiments.
REFERENCES Brailsford, A. D. and Major, K. G. (1964) The thermal conductivity of aggregates of several phases, including porous materials. Br. J. uppl. Phys. 15, 3 13. Brownlee, D. (1989) The composition of cometary nuclei. Proceedings of an International Workshop on Physics and Mechanics of Cometary Materials, pp. 233-236. ESA SP302. Espinasse, S. (1989) Modelisation du comportement thermique et de la differentiation chimique des noyaux de cometes. Dissertation, Laboratoire de Glaciologie et Geophysique et de l’Environment, Grenoble. Fanale, F. P. and Salvail, J. R. (1989) The water regime of asteroid (1) Ceres. Icarus 82,97. Fanale, F. P., Salvail, J. R., Zent, A. P. and Postawko, S. E. (1986) Global distribution and migration of subsurface ice on Mars. Icarus 67, 1. Greenberg, J. M. (1986) Evidence for the pristine nature of comet Halley. Proceedings of an ESA workshop on Comet Nucleus Sample Return, pp. 47-55. ESA SP-249. Griin, E., Benkhoff, J., Bischoff, A., Diiren, H., Hellmann, H., Hesselbarth, P., Hsiung, P., Keller, H. U., Klinger, J., Kniilker, J., Kochan, H., Neukum, G., ohler, A., RBssler, R., Spohn, T., Stiiffler, D. and Thiel, K. (1989) Modifications of comet materials by the sublimation process. Results from simulation experiments. Proceedings of the Workshop Analysis of Returned Comet Nucleus Samples. Milpitas, 1618 January 1989. Herman, G. and Podolak, M. (1985) Numerical simulation of comet nuclei, I. Water-ice comets. Icarus 61,252. Jeffrey, D. J. (1973) Conduction through a random suspension of spheres. Proc. R. Sot. Land. Ser A, 335.
water ice at low gas pressures
513
Klinger, J. (1981) Some consequences of a phase transition of water ice on the heat balance of comet nuclei. Icarus 47, 320. Knudsen, M. (1946) The Kinetic Theory of Gases. Methuen, London. Kiimle, N. I., Steiner, G., Dankert, C., Dettleff, G., Hellmann, H., Kochan, H., Baguhl, M., Kohl, H., KBlzer, G., Thiel, K. and Ohler, A. (1991) Ice sublimation below artificial crusts: results from comet simulation experiments. Planet. Space Sci. 39, 515. Ktihrt, E. (1984) Temperature profiles and thermal stresses in cometary nuclei. Icarus 60; 512. Linskv. J. (1966) Models of the lunar surface including temperature‘dependent thermal properties. Icarus 5,606. Parrott, J. E. and Stuckes, A. D. (I 975) Thermal Conductivity of Solids. Pion, London. Smoluchowski, R. (1982) Heat transport in porous cometary nuclei. J. geophys. Res. 87, A422. Spohn, T. and Benkhoff, J. (1990) Thermal history models -for KOSI sublimation experiments. Icarus 87, 358. Sauvres. S. W.. McCav. C. P. and Revnolds. R. T. (1985) *Temperatures within ‘comet nuclei. 3. geophys. Res. 961 12381. Steiner, G. (1991) Two considerations concerning the free molecular flow of gases in porous ices. Astron. Astrophys. (in press). Tschudin, K. (1946) Rate of vaporization of ice. Helv. Phys. Acta 19,91. Tsotsas, E. and Martin, H. (1987) Thermal conductivity of packed beds: a review. Chem.-Ing. Process 22, 19. Zchner, P. and Schliinder, E. U. (1970) Wlrmeleitfahigkeit von Schilttungen bei mlssigen Temperaturen. Chem.-Zng. Technol. 42,933. Zehner, P. and Schliinder. E. U. (1972) Einfluss der Wlrmestrahlung und des Druckes auf den Wlnnetransport nicht durchstriimter Schiittungen. Chem.-Zng. Technol. 44, 1303. Zucrow, M. S. and Hoffman, J. D. (1976) Gus Dynamics, Vol. I. Wiley, New York.
Space Sci., Vol. 39, No. 3, pp. 507-513, in Great Britain.
A MODEL POROUS
00324633/91 03.00f0.00 Pergamon Press plc
1991
OF THE THERMAL CONDUCTIVITY OF WATER ICE AT LOW GAS PRESSURES GERHARD
Institut
fur Weltraumforschung,
STEINER and NORBERT I. K6MLE
iisterreichische Akademie der Wissenschaften, A-8042 Graz. Austria
Observatorium
Lustbiihel,
(Received in$nal form 5 September 1990) Abstract-This study is concerned with energy transfer in porous water ice at large Knudsen numbers of the sublimated gas. We present a formula for the effective thermal conductivity, &s, which is found to be valid for a wide range of material parameters, in particular also for high porosity materials, where other expressions for &published in the literature lose their validity. It is found that above m 190 K &increases strongly with temperature and depends mainly on the average grain size, while at low temperatures it is primarily controlled by the value of the Hertz-factor. The influence of the porosity is relatively weak. The reliability of our expression for I,, has been checked by comparing computed temperature profiles with the temperature recordings of comet simulation experiments. The predicted temperatures are in good agreement with the experimental results. Finally, our model is applied to a typical surface layer of a cometary nucleus. A strong temperature dependence of the effective thermal conductivity in cometary water ice is predicted.
1. INTRODUCTION
Water
ice is present
exists
below
in comet
the surface
nuclei
of Mars
and probably and
minor
In all studies where the ice is assumed to be porous it is pointed out that the vapour phase significantly contributes to the energy transfer. Here we discuss the problem of energy transfer in porous water ice within the scope of laboratory experiments. In Section 2 we present a new approach to calculate the effective thermal conductivity of porous water ice at low gas pressures which is based on the work of Zehner and Schliinder (1970). In order to demonstrate the validity of the model it is applied to comet simulation (KOSI) experiments (Section 3). In Section 4 the dependence of Izelfon the material parameters is discussed, and the relevance of our results for the thermal properties of cometary surface layers is noted briefly.
also planets
(Fanale et al., 1986, 1989). For comet Halley H,O was found to comprise more than 80% of the volatile outflow, and there is evidence that water is the dominant volatile in most observed comets (Brownlee, 1989). Although coma observations do not directly relate to the composition of the nucleus, sublimation of water ice and heat transfer into its interior is thought to be the driving mechanism for the evolution of cometary nuclei. Hence, insight into the physical processes related to the transfer of energy in water ice is a necessary precondition for reliable model calculations. This fact is reflected in the new generation of comet models which largely depend on the physical properties of water ice. In a number of studies related to the evolution of comets the thermal properties of compact water ice are used to compute temperature profiles in the cometary interior &linger, 198 1; Kiihrt, 1984; Herman and Podolak, 1985). Smoluchowski (1982) first took into account that comet nuclei are probably highly porous bodies and derived an expression for the effective thermal conductivity of a porous icy nucleus utilizing the classical Maxwell equation. Squyres et al. (1985) computed temperatures within comet nuclei using an equation first given by Brailsford and Major (1964) for the effective thermal conductivity of the material. A different approach was taken by Espinasse (1989) and Spohn and Benkhoff (1990), who considered a model of coupled heat and mass transfer within an ice matrix.
2. HEAT CONDUCTION
MODEL
The one-dimensional, time-dependent heat conduction equation in a porous body with temperaturedependent material properties is given by
Here $ is the porosity of the material, pice is the mass density of compact ice, ci, is the heat capacity of compact ice, 1,s is the effective thermal conductivity, T is temperature, x is the depth below the ice surface, and t is time. Equation (1) can be solved for T with appropriate boundary and initial conditions, provided that a suitable expression for I,, is available. In the following we will derive an expression for I,e at low gas pressures by treating porous water ice without 507
508
G. STEINER and
N. I. KOMLE
admixture of dust or other volatiles as a two-phase material, where the two phases are the ice grains and the void spaces. 2.1. Thermal conductivities of the single phases In the SI system the thermal conductivity of crystalline water ice is given by (Klinger, 1981) li,
= 567/T.
‘E
(2)
Energy transport in the voids occurs via thermal radiation and the sublimated gas. At large Knudsen numof a void space, bers lvold, the thermal conductivity can be derived as follows : consider two planes of ice separated by a distance S and kept at temperatures T and T+AT, respectively. The energy removed from a plane at temperature T per unit area and unit time is given by
FIG. 1. THERMAL AND
CONDUCTMTY OF COMPACT ICE (CURVE 1) OF A VOID SPACE FOR ,'$= lo-' Ill(CURVE 2) AS A FUNCTIONOFTEMPERATURE.
Curves 3 and 4 are the contributions of thermal radiation and the transfer of latent heat to AVoid,respectively.
l/Z
P(2kT+L’).
(3)
Here E is the infrared emissivity of the ice, Q is the Stefan-Boltzmann constant, k is the Boltzmann constant, m is the mass of a gas molecule, L’ is the latent heat of sublimation per molecule, and P is the vapour pressure given by the ClausiusClapeyron equation P = a embiT,
(4)
where a and b are two experimental parameters (see Fanale et al., 1986). In writing down equation (3) it was assumed that the sublimation of H,O ice into vacuum can be described by the Hertz-Knudsen formula. This assumption has been experimentally verified by Tschudin (1946). The first term in equation (3) is the emitted thermal radiation, the second term comprises the energy carried by the gas molecules, where 2kT denotes their mean thermal energy (Knudsen, 1946). For AT << T*/b, the energy flux between the planes can be linearized around T. Consideration of the leading contributions and comparison with the Fourier heat conduction equation gives AvoIdas
“* dP dTLS,
(5)
where E’ is the infrared emissivity corrected for multiple reflections and L denotes the latent heat of sublimation per unit mass. Note that the contribution of the transfer of internal energy to the energy flux, both by conduction and advection, is negligible compared with the latent heat effect. For other geometries of the void different geometric factors have to be inserted into equation (5) (see Steiner, 1991). However, their influence on Avoidis not significant and will not be
further discussed here since, as will be seen, we will express I,, as a parametrized equation. In order to illustrate the relative importance of li, and AvoidFig. 1 shows the thermal conductivities of the single phases as a function of temperature for S = lo- 3 m. 2.2. Effective thermal conductivity Many different relations have been suggested in the literature to determine the effective thermal conductivity of a two-phase material. A classical formula is due to Maxwell, who derived 1,, for a dilute suspension of spheres of conductivity 1, embedded in a continuum of conductivity 12*by solving the Laplace equation for temperature around one sphere. The resulting expression for leff is first order accurate in tj, and hence only correct in the limit $ -+ 0. Notwithstanding this restriction many workers have used Maxwell’s result as though it were valid for all porosities. However, at high porosities it is not clear which phase is to be taken as continuous, and l,e depends strongly on this choice. This is demonstrated in Fig. 2, which shows leff as a function of temperature for $ = 0.5 and S = lo-’ m. In curve 1 the ice phase was chosen as continuous, and in curve 2 the void spaces were taken as the continuous phase. It can be seen from the figure that in the investigated temperature range & closely follows the thermal conductivity of the continuous phase. This observation provides a natural explanation for the circumstance that the effective thermal conductivity of porous water ice as given by Smoluchowski (1982), who utilized Maxwell’s equation with the ice phase as the continuous medium, is not suitable to explain the temperature curves recorded in comet simulation experiments in the big space simulator (see Spohn and Benkhoff, 1990). Note that a similar argument as to
509
Thermal conductivity of porous water ice at low gas pressures
water ice at large Knudsen numbers of the sublimated vapour can be written as &I = (1 -X/=&%~d+%,/=?
B is the “deformation
factor” which controls particle shape and is related to porosity via .
T
(ii1
FIG. 2. EFFECTIWTHERMALCONDUCTIVITYOFPOROUSWATER ICE AS A FUNCTION OF TEMPERATURE CALCULATED FROM MAXWELL'SEQUATION. In curve I the ice is considered as the continuous phase ; in
curve 2 the void spaces are considered
as the ~ntinuous phase. Also shown are ,I,, (curve 3) and ,I,,, (curve 4).
Maxwell’s equation applies to Russel’s equation, which yields similar results as Maxwell’s equation, and was used by Espinasse (1989). An updated version of Maxwell% approach has been obtained by Jeffrey (1973) who derived an expression for &.s which is second order accurate in $. However, this method also only applies to small porosities. For high porosity materials the effective thermal conductivity can be derived by simulating the thermal conductivity by a combination of parallel and series connections of “thermal resistances”. The simplest approach is to either average conductivities or resistivities or use a weighted sum of parallel and series connections (see Parrot and Stuckes, 1975; Tsotsas and Martin, 1987). An elaborate model to calculate the effective thermal conductivity of a system of particles in contact surrounded by a fluid, which has been tested against a wide set of data, was worked out by Zehner and Schltinder (Tsotsas and Martin, 1987 ; Zehner and Schliinder, 1970, 1972). They consider a unit cell characteristic for the material, which consists of an outer fraction att~buted to the fluid phase only and a core covered by both the solid and the fluid phase, and derive &t by an electric circuit analogy as indicated above. Their model accounts for heat conduction in the fluid phase, the solid phase, heat transfer over particle contact points and thermal radiation. An expression for a, in porous water ice at low gas pressures can be obtained from their model [see Tsotsas and Martin, 1987 ; equation (S)] by substituting lLra,,,the radiative thermal conductivity of a void space, by lvold and taking the limit L + 0, where 1 denotes the thermal conductivity of the fluid phase. Thus, the effective thermal conductivity of porous
(h is the so-called flattening coefficient. It determines the amount of heat flowing through particle contact points and is equivalent to a Hertz-factor which accounts for the reduced heat &IX in granular material compared with compact material. Note that in the derivation of equation (6), S, the average dimension of a pore space, has been replaced by $d, where d denotes the average particle diameter (Zehner and Schlunder, 1972). Thus, d is the relevant length scale which has to be used in dete~ining ;iyoid in the expression for &. Equation (6) consists of three terms that contribute to the effective thermal conductivity of the material: the first term accounts for energy transfer in the void spaces, the second term for heat transfer over particle contacts, and the third term for the energy transferred in the core of the unit cell which is covered by both the fluid and the solid phase. A wide range of flexibility to determine & for different materials is provided by three adjustable parameters that characterize the investigated substance. These parameters are the flattening coefficient 4, the average particle size d, and the porosity 3/. 3. APPLICATION
TO COMET
SIMULATION
EXPERIMENTS
Recently the thermal evolution of ice samples under space conditions has been studied by laboratory experiments in two space simulators at the DLR/Koln (F.R.G.) (for a review see Griin et al., 1989). In the following we report about a series of experiments in the small vacuum chamber carried out within the framework of the comet simulation (KOSI) programme. The main purpose of these experiments was to study the influence of a “porous crust” on the thermal evolution of the ice sample. 3.1. Experimental description The sample container was a 10 cm sized box with Teflon walls and a copper back plate. The samples consisted of frozen droplets of water ice, were covered
G.
510
and N. I. K~MLE
STEINER
Y
T [I<) FIG. FREE
3. KNUDSEN PATH
TEMPERATURE
OF
NUMBER THE
FOR
GAS
s =
S/S, WHERE MOLECULES,
lo-’ m
6 DENOTES AS
CALCULATED
A
THE MEAN
FUNCTION FROM
OF
KINETIC
GAS THEORY.
The vapour pressure was calculated from equation (4). The region between the dashed lines is the transition region between the free molecular and continuum flow regime as given by Zucrow and Hoffman (1976).
FIG.
4a.
COMPUTED
OVER A TIME PERIOD
CIllBELOW WITH
THE
TEMPERATURE OF
4h
ICE SURFACE
THE MEASURED
3.2. Model calculations Now we report the computations for one experiment which is taken as representative for the whole series. Equation (1) has been solved for temperature with Aerrexpressed by equation (6). The applicability of equation (6) is demonstrated in Fig. 3, where the Knudsen number of sublimated water vapour is shown as a function of temperature for S = lo- 3 m. Note that the plotted curve provides a lower limit for the Knudsen numbers occurring in the experiments. The temperature of the ice surface (upper boundary) was approximated by
TO
OF 1,
(SOLID
LINES)
2, 3, 5
AND
BOTTOM)
cuRvEs
COMET SIMULATION
T, = Ti+(T,‘-ps) by steel plates containing an array of equidistant holes simulating a porous crust, and were irradiated by an artificial sun. The chamber pressure was kept at x lo- * Pa. The back plate of the sample container was cooled to - 120 K. The ice temperature was recorded over 5-6 h at a depth of 1,2, 3, 5, and 7 cm below the ice surface. The main effect of the crust was to raise the surface temperature, depending on the porosity of the crust, in comparison with a freely sublimating ice surface. At the uppermost temperature sensor a constant temperature was reached after -30 min. After 4 h the samples were nearly isothermal down to more than 75% of their depth. The temperature curves within the samples attained the convex shape that has also been observed during experiments in the big space simulator (Spohn and Benkhoff, 1990). The porosity of the sample material was determined as $ x 0.5, the average grain size was estimated to lie in the range of 0.1-I mm. The experiments are described in more detail in KGmle et al. (1991).
(TOP
TEMPERATURE
FOR A REPRESENTATIVE
EVOLUTION
AT A DEPTH
tanh(t/r),
7
COMPARED
(DASHED
LINES)
EXPERIMENT.
(8)
where TS is the initial surface temperature which is obtained by linear extrapolation from the initial temperature below 1 cm depth, T,’ is the temperature at the uppermost temperature sensor after a steady state has been reached, and z is a time constant typical for the warming up period of the surface. At the lower boundary the measured temperature of the back plate has been used. The initial temperature profile has been produced by linear interpolation between the temperatures determined at the positions of the temperature sensors. Equation (1) has been solved for temperature by an implicit numerical code. The result was checked by recalculating the solutions with an explicit finite difference method. The porosity was taken as $ = 0.5, which is in agreement with the values determined experimentally. The adjustable parameters were chosenas~=4~10~~,d=O.l6mmandr= 16min. This choice gave the best fit to the recorded temperatures. It should be noted that for the other experiments the adjustable parameters vary at most by a factor of three. Figure 4a shows the computed temperature evolution (solid lines) over a time period of 4 h at a depth of 1, 2, 3, 5 and 7 cm below the ice surface in comparison with the measured temperature profiles (dashed lines). The accuracy of the computed temperatures is within 10% of the observed temperatures. A systematic deviation from the recorded temperatures occurs at the two lowermost temperature sensors. This feature is observed in all experiments and is speculatively attributed to radial heat fluxes which are not considered in the model. Figure
Thermal
conductivity
of porous
water ice at low gas pressures 100
FIG. 4b. COMPUTED SPATIAL TEMPERATURE DISTRIBUTION IN THE ICE SAMPLE CORRESPONDING TO FIG. 4a AFTER 0.5,1, 1.5, 2,3 AND 4 hINTOTHEEXPERImNT (LERTORIGHT). The measured temperatures are indicated by dots.
FIG. 6. INFLUENCE OF THE AVERAGE PARTICLE SIZEON &AS PREDICTED BY EQUATION (6). 4 = lo-*, ((I =0.5 and 10m6 m < d-c 10m2 m (bottom to top). d is increased by a factor of 10 between two neigh-
bouring
curves.
resulting curves for the effective thermal are in good agreement in the investigated range and for the utilized parameters. 4. DISCUSSION
T
IKI
FIG. ~.EFFECTI~ETHERMALCONDUCTIVITYOFTHEICESAMPLE PERTAINING TO FIG. 4a AND b COMPUTED FROM EQUATION (6) (SOLID LINE).
Also shown are the curves for & calculated from the equation of Linsky (dashed dotted line) and Brailsford and Major (dashed line).
4b shows the spatial temperature profiles for the same experiment at z = 0.5, 1, 1.5, 2, 3 and 4 h computed from equation (1) (solid lines) compared with the measured temperatures (dots). Figure 5 shows the thermal conductivity of the ice sample pertaining to Figs 4a and b computed from equation (6). It is seen that at low temperatures the derived leff- closely follows c#&. At temperatures higher than approximately 190 K it strongly increases with temperature and eventually exceeds lice. Also shown in Fig. 5 are the curves computed after Linsky (1966) whose method to calculate 1, (adopted to the current problem) consists of simply adding &, and &,id, and the equation of Bra&ford and Major (1964), again with the thermal conductivity of the solid phase taken as $J&. As can be seen from the figure the
conductivity temperature
AND CONCLUSIONS
We have derived an expression for the effective thermal conductivity of porous water ice at low gas pressures and utilized it to reproduce the temperatures recorded in comet simulation experiments. Now it is interesting to discuss in more detail the influence of the adjustable parameters, 4, d and r,G, which characterize the investigated material, on the effective thermal conductivity according to equation (6). The effect of the average particle size d is shown in Fig. 6, where &is displayed for 10m6m < d < lo-* m and fixed values of 4 and +. It is seen that the average particle size significantly affects 1,, at temperatures higher than approximately 170-210 K. At low temperatures 1,, is mainly controlled by the value of the flattening coefficient since the conductivity of the voids is small at low temperatures. This is displayed in Fig. 7, where &is plotted for fixed d and $ and lo- 5 < r$ < 1. From Fig. 8, where lefl is shown for fixed d and 4 and two extreme values for the porosity, it is obvious that the effect of porosity on the effective thermal conductivity is small compared with the influence of d and 4. Note that while the influence of the average particle size and the flattening coefficient on l,= as predicted by equation (6) is in good agreement with the corresponding predictions from the equations of Linsky and Brailsford and Major, different conclusions are drawn concerning the influence of porosity. Linsky’s expression on the one hand is independent of $, the
G. STEINER and
N. I. K~~MLE
000,
’
1111111(111
T CKI FIG. ~.INFLUBNCEOFTHEFLATTENING COEFFIC~ENTON,&AS PREDICTED BY EQUATION (6).
d= 10e3 m, + = 0.5and lo-’ < C#J< 1 (bottom to top). C/J is increased by a factor of 10 between two neighbouring CWWS.
FIG.
8.
INFLUENCE OF POROSITY ON lcffAS PREDICTED BY EQUATION (6). d = lo-' m, 4 = lo-‘, $I = 0.9 (solid line) and + = 0.1
(dashed line). equation of Brailsford and Major on the other hand exhibits a similar behaviour to Maxwell’s equation at high and low porosities. Thus, it predicts that at high volume fractions of the void spaces 1,, tends to zero with Avoidat low temperatures (see Fig. 9). This behaviour is in contradiction to the observation that individual grains may form high porosity agglomerates of particles in contact over whose contact areas heat flow can occur so that 1,s is expected to be proportional to 4iice at low temperatures. This line of evidence supports the validity of equation (6), and we will use it to discuss briefly a possible scenario for the thermal conductivity of water ice present in comets. Greenberg has proposed the possibility that mostly sub-micrometre-sized dust grains covered by refractory organics and volatiles constitute the building elements of comets. The size of these composite particles is suggested to lie in the micrometre range (see
FIG. 9. EFFECTIVETHHWALCONDUCT~VITYASAFUNCTIONOF TEMPERATURE COMPUTED FROM EQUATION (6) (SOLID LINE), THE EQUATION OF LINSKY (DASHED DOTTED LINE) AND THE EQUATION OF BRAIL~FORD AND MAJOR (DASHED LINE) FOR d = 10m3 m, cj~= lo-’ AND + = 0.9.
FIG. 10. POSSIBLE THERMAL CONDUCTIVITY OF THE NEAR SURFACEREGIONOFACOMETARYNUCLEUS(SOLID LINE).
Also shown is the thermal conductivity (dashed line).
of compact ice
Greenberg, 1986). The value of the Hertz-factor can be expected to be much smaller in the cometary situation (probably below 10e5) than was found in our experiments, which were performed in a one-genvironment. Figure 10 displays the thermal conductivity of porous water ice as it may be typical for the near surface layer of a cometary nucleus, which is already depleted of the more volatile species. The effect of a probable admixture of dust particles is not considered here. The parameters chosen are d = 10m6 m, $ = IO-’ and $ = 0.8. Also shown in Fig. 10 is the thermal conductivity of compact water ice in the same temperature range. It is clear from the figure that l,e of such a material is completely different from li,. At low temperatures it is proportional to 4&c ; at temperatures higher than approx. 190 K it increases with temperature and varies by about three orders of magnitude in the relevant temperature range.
Thermal
conductivity
of porous
Acknowledgements-This work has been supported by the Austrian “Fonds zur Fijrderung der wissenschaftlichen Forschung” under grant number P7246-GEO. Furthermore we wish to thank the members of the KOSI team for providing the opportunity to perform the experiments.
REFERENCES Brailsford, A. D. and Major, K. G. (1964) The thermal conductivity of aggregates of several phases, including porous materials. Br. J. uppl. Phys. 15, 3 13. Brownlee, D. (1989) The composition of cometary nuclei. Proceedings of an International Workshop on Physics and Mechanics of Cometary Materials, pp. 233-236. ESA SP302. Espinasse, S. (1989) Modelisation du comportement thermique et de la differentiation chimique des noyaux de cometes. Dissertation, Laboratoire de Glaciologie et Geophysique et de l’Environment, Grenoble. Fanale, F. P. and Salvail, J. R. (1989) The water regime of asteroid (1) Ceres. Icarus 82,97. Fanale, F. P., Salvail, J. R., Zent, A. P. and Postawko, S. E. (1986) Global distribution and migration of subsurface ice on Mars. Icarus 67, 1. Greenberg, J. M. (1986) Evidence for the pristine nature of comet Halley. Proceedings of an ESA workshop on Comet Nucleus Sample Return, pp. 47-55. ESA SP-249. Griin, E., Benkhoff, J., Bischoff, A., Diiren, H., Hellmann, H., Hesselbarth, P., Hsiung, P., Keller, H. U., Klinger, J., Kniilker, J., Kochan, H., Neukum, G., ohler, A., RBssler, R., Spohn, T., Stiiffler, D. and Thiel, K. (1989) Modifications of comet materials by the sublimation process. Results from simulation experiments. Proceedings of the Workshop Analysis of Returned Comet Nucleus Samples. Milpitas, 1618 January 1989. Herman, G. and Podolak, M. (1985) Numerical simulation of comet nuclei, I. Water-ice comets. Icarus 61,252. Jeffrey, D. J. (1973) Conduction through a random suspension of spheres. Proc. R. Sot. Land. Ser A, 335.
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513
Klinger, J. (1981) Some consequences of a phase transition of water ice on the heat balance of comet nuclei. Icarus 47, 320. Knudsen, M. (1946) The Kinetic Theory of Gases. Methuen, London. Kiimle, N. I., Steiner, G., Dankert, C., Dettleff, G., Hellmann, H., Kochan, H., Baguhl, M., Kohl, H., KBlzer, G., Thiel, K. and Ohler, A. (1991) Ice sublimation below artificial crusts: results from comet simulation experiments. Planet. Space Sci. 39, 515. Ktihrt, E. (1984) Temperature profiles and thermal stresses in cometary nuclei. Icarus 60; 512. Linskv. J. (1966) Models of the lunar surface including temperature‘dependent thermal properties. Icarus 5,606. Parrott, J. E. and Stuckes, A. D. (I 975) Thermal Conductivity of Solids. Pion, London. Smoluchowski, R. (1982) Heat transport in porous cometary nuclei. J. geophys. Res. 87, A422. Spohn, T. and Benkhoff, J. (1990) Thermal history models -for KOSI sublimation experiments. Icarus 87, 358. Sauvres. S. W.. McCav. C. P. and Revnolds. R. T. (1985) *Temperatures within ‘comet nuclei. 3. geophys. Res. 961 12381. Steiner, G. (1991) Two considerations concerning the free molecular flow of gases in porous ices. Astron. Astrophys. (in press). Tschudin, K. (1946) Rate of vaporization of ice. Helv. Phys. Acta 19,91. Tsotsas, E. and Martin, H. (1987) Thermal conductivity of packed beds: a review. Chem.-Ing. Process 22, 19. Zchner, P. and Schliinder, E. U. (1970) Wlrmeleitfahigkeit von Schilttungen bei mlssigen Temperaturen. Chem.-Zng. Technol. 42,933. Zehner, P. and Schliinder. E. U. (1972) Einfluss der Wlrmestrahlung und des Druckes auf den Wlnnetransport nicht durchstriimter Schiittungen. Chem.-Zng. Technol. 44, 1303. Zucrow, M. S. and Hoffman, J. D. (1976) Gus Dynamics, Vol. I. Wiley, New York.