A value is described for the thermal conductivity of the frost layer and for the water-ice solid debris mixture. The value of the porous structure is discussed as a function of depth only. Graphs show thermal conductivity as a function of depth and temperature at constant porosity and density

Khader, M. S.

Kumar, G. N.

Vachon, R. I.

English

NASA Technical Reports Server

AUBURN UNIVERSITY
THERMAL CONDUCTIVITY OF COMETS
Final Report
Contract NAS8-26579
by
R. I. Vachon, G. N. Kumar, M. S. Khader
School of Engineering
Department of Mechanical Engineering
Thermoscience Group
Under Contract With
George C. Marshall Space Flight Center
National Aeronautics & Space Administration
Administered Through
ENGINEERING EXPERIMENT STATION N
AUBURN UNIVERSITY
Auburn, Alabama 36830
September 1974
NASA-CR-120524) THERMAL CONDUCTIVITY OF N75-1180
5
COMETS Final Report (Auburn Univ.)
26 p BC $3.75 CSCL 03B UnclasUnclas
G3/89 02582
https://ntrs.nasa.gov/search.jsp?R=19750003733 2020-03-23T03:00:18+00:00Z
NOMENCLATURE
B A constant in Eqs. (2) and (4)
C A constant in Eqs. (2) and (4)
C' Negative of C
K Thermal conductivity (WATT/cm-ok)
P Phase volume fraction
Pl Porosity
P Mass fraction
m
R Thermal resistance
T Absolute temperature (ok)
E Modulus of Elasticity of solid particle
g Gravitational constant
z Depth below the surface
p Density
p Average density
u Poisson's ratio
Subscripts
c Continuous phase
d Discontinuous phase
eff Effective
S Solid
i
INTRODUCTION
A comet is generally regarded to be composed of three primary regions:
the nucleus or Kernal, the coma (a plasma region surrounding the nucleus),
and the tail. Since not much is known about the composition and structure
of the nucleus, it is thought that some useful information may be inferred
by investigating the heat and mass transfer of the nucleus. Before con-
ducting these investigations, it is necessary to study the material thermal
properties of possible nucleus models.
The object of the present study program is to perform the following:
Part 1: To recommend a value for the thermal conductivity of the frost
layer and a value for the thermal conductivity of the water-ice
solid debris mixture. The basis for the recommendation is a
nucleus model consisting of water-ice mixed with basaltic and
meteoritic materials in the form of dust and agglomerated par-
ticulates, all of which is surrounded by a layer of water-frost,
assuming a constant thermal conductivity in each of the layers.
Part II: To recommend a value for the thermal conductivity of the porous
structure of Part II as a function of depth only. The model for
this part is the same model as in Part I with the following addi-
tional assumption. The assumption is that the frost layer and
water-ice have been sublimating and evaporating, leaving a porous
structure composed of the solid basaltic and meteoritic material
with residual gases partially confined in the porous structure to
varying amounts over some depth. This conductivity function should
be regarded as a simplified approximation to be used for initial
calculational estimates of heat transfer in the nucleus.
1
2MODEL FOR PART I
WATER FROST
WATER ICE +
BASALT + METEORIT
DEPTH * MTR
FIGURE 1 (a).
MODEL FOR PART II
BASALT + METEORITIC
MATERIAL + RESIDUAL
GASES.
DEP TH--
FIGURE 1(b)
3COMETS CONSIDERED
i) Commet Encke: With a minimum temperature of 1400 K, Radius: 4KM.
ii) Comet Halley: Min. Temp: 49'K, Radius: 15KM.
ANALYSIS FOR PART I
For the Water Ice-Solid Debris Mixture, the Cheng and Vachon Model [1]
has been used to calculate the approximate effective thermal conductivity
of this layer. The nucleus of a comet normally consists of 70% or more
of ices by mass [2] and the rest, basaltic and meteoritic material. As a
first approximation, in the absence of detailed information on the composition
of the meteoritic material (other than that it might contain Fe, Ca, Mn,
Mg, Cr, Si, Ni, Al, etc), the mixture of basaltic and meteoritic material
has been approximated first with the properties of basalt and then with
properties of basalt and iron which are mixed in different proportions.
(A): When the solid debris has been approximated with the properties
of basalt, the above model is simplified to a mixture of water ice and
basaltic material. Since the temperature of the nucleus varies with its
distance from the Sun (Example: At Aphelion [3] temperature of nucleus
of Encke = 1420 K and that of Halley = 490 K),the thermal conductivities
have been calculated for the various assumed temperatures of the nucleus.
The calculations have also been performed with varying proportions of
ice and basaltic material.
(B): When the solid debris was approximated by a mixture of basaltic
material and iron, the model is simplified to a mixture of ice, basaltic
material and iron. Again the calculations have been performed for different
assumed temperatures of the nucleus (variation of thermal properties within
the nucleus for a given average temperature have not been considered) and
for varying proportions of water ice and solid debris. The calculations
4have also been performed with varying proportions of basalt and iron for
a given ice, solid debris ratio.
A linear variation of thermal conductivity with temperature has been
assumed for the water ice and using the values given in [4]. The follow-
ing equations were obtained for the thermal conductivity and the desnsity
variation of ice with temperature
-5a) K.ic e = [0.0481156 - 9.2444 x 10- T]Watt/cm-Kice
(1)
-5 3b) pice = [0.94323 - 9.2444 x 10- 5 T] gm/cm T in oK.
The following property values were used for the Basalt [5] and Iron [6].
Basalt: p = 2830 Kg/m 3  K = 1.34 watt/moK5
E = 2.2 x 10" N/m2  v = 0.2
Iron: p = 7890 Kg/m 3  K=63.7 watt/m°K.
Ss
E = 2.07 x 10" N/M2  v = 0.3
K. is of the order 3.8 watt/m°K at 100 0 Kice
I(A)
For this model, we see from the above that for the temperature
K.
ice
range of 30-2000 K, the ratio Kie is in the range 1 to 4. Hence,solid
from [1], for the case Kc > K we have the effective or equivalent thermal
resitance given by
R2 tan- 1 B -C'(Kd - Kc) 1-B
f - C (k d - k c ) [Kc + B (Kd Kc)] ) K + B(K - Kc) + K (2)c d c
Where K = Thermal conductivity of continuous phase
c (water-ice in this case)
K = Thermal conductivity of discontinuous phase
d (basalt in this case)
Pd = Discontinuous phase volume fraction
B = / 3P2d/2
C = -4/B
C' = -C
Then the approximate effective thermal conductivity is given by
1
eff - (3)
ef f
Thus, the Kef f has been calculated for different temperatures of
nucleus and for different phase volume fractions of the discontinuous
phase (basalt in this model).
I(B)
In this model, we have a three phase mixture of water-ice, basalt
and iron and we have to find the approximate thermal conductivity (Keff)
of this three-phase mixture. Cheng and Vachon [1] give the following
method for calculating Keff:
Since the major portion of the comet nucleus is made up of water-
ice, we take water-ice to be the continuous phase in applying the equation
developed by Cheng and Vachon for 3 phase mixtures.
Let K = Thermal conductivity of the continuous phase (i.e.
c of water ice)
K = Thermal conductivity of the first discontinuous
S phase (i.e. of Fe)
Kd = Thermal conductivity of the second discontinuous
2 phase (that is of basalt)
6Thus, from the given range of values for the above, we have
Kd Kd
> 10 -2 <1K K
c c
In order to determine the effective thermal conductivity (Keff)
of this 3 phase mixture, the 3 phase mixture is considered to be
reduced to a two-phase mixture in which the continuous phase is composed
of the original continuous phase of the three phase mixture and the
second discontinuous phase. The discontinuous phase of this two phase
mixture is the first discontinuous phase of the three-phase mixture.
For this two-phase mixture,
P = Pd1
P = P + P
ce c d
K = Kd
dd
K = Thermal conductivity of the effective continuous phase
ce
Pce = Phase volume fraction of the effective continuous phase
Kce'
, 
the effective thermal conductivity of the original continuous
phase mixed with the second discontinuous phase of the 3 phase mixture,
is determined as per the method given in Part I(A).
Kd
Now, since -- > 10, from ref [1], the effective resistance Ref f
ce
is given by
1
Reff = /{C'(K - K ) [K + B (K - K )]}
d ce ce 1 d1  ce
K K B /{C'(K - K )}
K + B ( d - ce) 1 d ce
In /{ ce + 2 (4)
{Kc + B (K - K } - B {CI(K - K)} + 1 - B (4)
ce 1  d ce 1 d ce 11 -12 K
ce
7where B1 = /3Pd /2
1
C' = -C = + 4/B1
and the effective thermal conductivity is then given by
K eff = 1
eff Ref (5)
eff
This calculation has been performed for different assumed temper-
atures of the nucleus (Here also, the thermal property variation inside
the nucleus has not been considered) and for different phase volume
fractions of the continuous phase. For a given value of phase volume
fraction of water ice, calculations have been performed for varying
proportions of basalt and iron.
ANALYSIS FOR PART II
For the model in Part I, we now assume that the frost layer and
the water ice has been sublimating and evaporating which occurs as the
comet nears the sun (distance less than 2 AU), thus leaving a porous
structure.
A simplified approximation for the effective thermal conductivity
as a function of depth is obtained using the theoretical model developed
by Khader and Vachon [5] for heterogeneous mixtures. In the application
of the above model to our case, it has been assumed that since the gas
is at low pressure the thermal conductivity of the residual gases is
quite small compared to the thermal conductivity of basalt and meteoritic
material. Hencethe thermal conductivity through the void can be
neglected. Also, since the temperatures involved are of the order of
100 0K, radiation through the void space (i.e. through the gases at very
8low pressures) could be neglected as being very small in comparison to
conduction through the solid.
I I (A)
Thus, with these simplifications, from Ref [5] we get the following
expression for the effective thermal conductivity when the solid is
approximated with the properties of basalt.
2 Z2/3)
K = 3(1 - P) (1 - C 2 2 ) K
eff 1 s
7T 3 22 (2 c1/3 1) + (4- 1.2 C1 Z2/3)] (6)
2 cZ
2 1/3
where C, = 2 g pT(i-v )
16 E (1 - P,)
p = Average Density of the Solid.
P,= Porosity of the solid material.
The calculations have been performed for different porosities and
for different densities up to a depth of 2500 meters.
II(B)
When the solid material is approximated by a heterogeneous mixture
of basalt and Fe, the problem is solved in 2 stages. First, an effective
value of thermal conductivity is obtained for the heterogeneous mixture
of basalt and Fe using the Cheng and Vachon model [l].
Then the model for the second stage consists of the solid material
whose thermal conductivity was determined in Stage 1 and whose voids are
assumed to be filled with low pressure residual gases. The effective
9thermal conductivity of this stage is determined using the Khader and
Vachon model as explained for II(A).
10
RESULTS AND DISCUSSION: PART I
Part 1(A):
Fig. 2 shows the variation of effective thermal conductivity with the
average temperature when the solid debris is approximated with the
properties of basalt for different values of the density of the dis-
continuous phase at a given value of mass fraction of the discontinuous
phase [For example: The above graph is for the case when there is 80%
of water-ice and 20% of basalt by mass]. This figure shows that the
effective thermal conductivity of this layer is of the same order as
that of water-ice and that its variation with temperature is also linear.
The reason for this is that the major portion of the nucleus (about 70-80%)
is made up of water-ice and the thermal conductivity of basalt is also
of the same order of magnitude as that of water-ice. The figure also shows
that at a given temperature there is not much variation in Keff with change
in density for this case,especially near temperatures of the order of 2000 K.
Fig. 3 shows the variation of Kef f with mass fraction of solid debris
(i.e., discontinuous phase approximated by basalt in this case). It shows
that the variation is not appreciable in the range of mass fraction pre-
dicted for the nucleus of a comet and that there is a slight decrease in
the thermal conductivity as mass fraction of solid debris is increased.
It also shows that the variation is more pronounced at temperatures near
40 0K than near 2000K where it is almost flat.
Part I(B):
For the model of I(B), variation of thermal conductivity with the
total mass fraction of basalt and iron is shown in Fig. 4 for different
proportions of Fe and basalt. As can be seen from this figure, though
the curves start at almost the same value of Kef f for very low values of
mass fraction (the reason for which is that practically the whole nucleus
in this case will be made up of water-ice), the slope of the curve changes
from being negative for the model with only basalt, to positive for the
model with only Fe as the solid debris material. The reason for the
positive slope in the latter case is that since the thermal conductivity
of iron is much greater than that of water-ice, the thermal conductivity
of the mixture increases as the percentage of Fe increases. When Fe and
basalt are in the ratio 1:1, even though the slope is positive, the
increase is not marked.
12
PART II
Figs. 5, 6, 7 show the plots of variation of the effective thermal
conductivity with depth for different values of porosity, each figure for
a particular proportion of Fe and basalt. They all exhibit the same
general trend of increase of Kef f with depth, the increase being sharper
during the earlier part and then becoming gradual. The reason for this
is that as the depth increases the contact area between the particles
increases due to the increase in loading and consequently smaller thermal
contact resistance and in turn thermal conductivity increases. As the
porosity increases, the amount of solid material per unit volume decreases
and hence thermal conductivity decreases with porosity.
A comparative study of Figs. 5, 6, 7 is made in Fig. 8. Comparing
curves 1 and 5 we see that the thermal conductivity increases by
a factor of about 75 when basalt is replaced by Fe. This marked increase
in Kef f with increase in percentage of Fe compared to basalt is because
of the very high value of thermal conductivity for Fe compared to basalt.
This figure also shows that the curve of Keff sharply rises for the first
few hundred meters and then becomes very gradual (almost flat). Thus after
about 500 meters, the effect of depth on thermal conductivity is not
pronounced. This is because after this depth, no appreciable increase in
contact area between the particles occurs with depth.
Fig. 9 shows the variation of effective thermal conductivity with
depth for the case II(A) [i.e. with only Basalt as solid material] for
the first 200 meters from the surface. This shows that even though the
graph still follows the trend of Figs. 5, 6, 7, actually the major portion
of the increase in the value of Keff takes place within a few meters from
the surface and afterwards the increase is very gradual. Since the same
13
pattern was followed for the other cases (for varying proportions of
basalt and Fe) they have not been plotted.
Fig.10 illustrates the variation of Kef f with density at differ-
ent depths when the solid material was approximated with the properties
of basalt. The increase in thermal conductivity with density is very
gradual and the reason for the increase in Keff with density is the
increased contact area between the particles which in turn is due to
increase in loading due to heavier mass/unit volume at a given porosity.
Fig. 11 also illustrates the same.
14
4.5
C14
4.0
U4o
3.5
4-4 = 2.8 gm/cm
4-4
. -- 2.3 gm/cm3
S = 1.7 gm/cm3
$=1. 3 gm/ cm3,H
. 3.0
4-1
0
2.5
a)
E44
2.0
I I I I 1 I
20 60 100 140 180 220
Temperature OK
FIGURE 2: EFFECTIVE THERMAL CONDUCTIVITYAS A FUNCTION OF TEMPERATURE
AT CONSTANT DENSITY FOR PART I (A).'
C11
o 4.5
o
4.0
T 40° K
4-4
44
S3.5
Tood00
U 3.0- T 150 ° K
T-= 200 K
4
a 2.5
4'-4
1 I I1
44i
0.1 0.2 0.3 0.4 0.5
Pm" (Mass Fraction)
t IGURE 3: EFFECTIVE TPER.MAL CONDUCTIVITY AS A. FUNCTION OF MASS T'RACTION OF
SOLID DEBRIS(BASALT 4- METEORITIC MATERIAL ) AT CONSTANT TEMPERI.T TIU E
FOR PART I(A).
16
0.6
'4-4
0Q) 1J
0.5 1
2J
03 --
3
0
4-' 
' '
0 .4)
6
.3
4-4
W 1 Water Ice + (Fe + Basalt) In Ratio 1:0 At T = 50 R
2 Water Ice + (Fe + Basalt) In Ratio 1:1 At T = 50'R
3 Water Ice + Basalt At T = 50°R
0.2- 4 (1) At T = 150 R
5 (2) At T = 150R
6 (3) At T = 15 0aR
0.1 -
I I I I I I
0.10 0.15 0.20 0.25 0.30 0.35
P = Mass Fraction Of (Fe + Basalt)
m
FIGURE 4: EFFECTIVE THERMAL CONDUCTIVITY AS A FUNCTION OF MASS FRACTION OF SOLID
DEBRIS (BASALT + METEORITIC MATERIAL ) AT CONSTANT TEMPERATURE FOR PART I(B).
17
0.8.
ONLY BASALT AS SOLID MATERIAL.
0.6 -
-4
P 0.70
P = 0.74
0.4 P = 0.78
S...-- 'P 
- 0.82
. -- P = 0.86
P = 0.90
) I I I I I
400 800 1200 1600 2000 2400
Depth Z in meters
FIGURE 5: EF'ECTIVE THERMAL C0 DUCTIVITY AS A FtNCTION OF DE.T, AT CONSTANT POROSTT
FOR PART IH(A).
18
0
14
u
-1. (IRON + BASALT ) AS SOLID MATERIALS IN TIHE RATIO 1:1.
. - -P= 0.70
4444 P= 0.74
P= 0.86P= 0.82-
" -- ... P= 0.90
4.4
0I I I I L I
Depth Z in meters
FIGURE 6: EFFECTIVE THERMAL CONDUCTIVITY AS- A FUNCTION OF.-DEPTH AT CONSTA-NT POROSITY
FOR PART II(B).
19
4.0-
IRON AS SOLID MATERIAL.
P=0.70
r=0.74
3.0-
F---RARP=0.78
P=0.82
2.0- -P=0.86
P=0.90
1.0
400 300 1200 1600 2000 2400
Z in meters
IGUE 7: EF'FECTIVE THERMAL CONDUCTIVITY AS A FUNCTION OF DEPTH AT CONSTANT POROSITY
FOR PART II(B).
20
5
10-2
.4A 444
-3
2
/ 1 With Only Basalt As The Solid Material
2 With Fe + Basalt In The Ratio 1:4
V 3 With Fe + Basalt In The Ratio 1:1
10 4 With Fe + Basalt In The Ratio 4:1
5 With Fe Only As Solid Material
11
POROSITY = 0.8
I I I I I I
400 800 1200 1600 2000 2400
Depth Z in meters
FIGURE 8: COMPARISION OF EFFECTIVE THERMAL CONDUCTIVITY FOR DIFFERENT PROPORTIONS
OF BASALT AND IRON AS A FUNCTION OF DEPTH AT CONSTANT POROSITY FOR PART II(B).
21
M
BASALT AS SOLID MATERIAL.
OfU
oI
4.)
0.4 -
4-4
S 0.3
.-
., P = 0.70
O
F-)
P = 0.76
0.2- P = 0.80
P = 0.86
u P = 0.92a) 0. -
4-i44
I I I I I
40 80 120 160 200
Depth Z in meters
.:IGURE 9: EFFECTIVE TNERMAL CONDUCTIVITY AS A FUNCTION OF DEPTH AT CONSTANT POROSITY
FOR PART II(A).
22
BASALT AS SOLID MATERIAL.
O
0
Z o500m
0.2-
Rz = S0.m
4.
I I I I I
1.4 1.8 2.2 2.6 3.0
DENSITY (gm /cm 3)
FIGURE 10: EFFECTIVE THERMAL CONDUC;TIVITY AS A FUNCTION OF DENSITY AT CONSTANT DEPTH
FOR PART II(A).
23
O BASALT AS SOLID MATERIAL.
O
POROSITY=0.8
4-4
4-4
0.6
3 = 2.83 gm/cm3
S= 2.30 gm/cm 3
=5 1.80 gm/cm3
3 = 1.40 gm/cm
0.4 -
0
a)
H0.2
4-4
II ... . ..
400 800 1200 1600 2000 2400
Depth Z in meters
FIGURE 11: EFFECTIVE THERMAL CONDUCTIVITY AS A FUNCTION OF DEPTH AT CONSTANT DENSITY
FOR PART II(A).
REFERENCES
(1) Cheng, S. C. and Vachon, R. I., "The Prediction of the Thermal
Conductivity of Two and Three Phase Solid Heterogeneous
Mixtures" Int. J. Heat Mass Transfer, Vol. 12, pp 249-264,
(1969).
(2) F. L. Whipple, "A Comet Model II - Physical Relations for Comets
and Meteors" Astrophysical Journal, Vol. 113, pp 464-474,
(1951)
(3) M.G.J. Minnaret, "On the Temperature of the Cometary Nucleii"
Proceeding of Academy of Sciences, A. M. Sterdam, #50,
pp 826, (1947).
(4) Lentz, C. P. "Thermal Conductivity of Meats, Fats, Gelatine, Gels
and Ice. Food Technology , 15, pp 243-244, 1961.
(5) M. S. Khader and R. I. Vachon, "A Theoretical Model for Lunar
Surface Material Thermal Conductivity". ASME Journal of Heat
Transfer, HT-35-1973.
(6) Eckert and Drake: Analysis of Heat & Mass Transfer. McGraw-Hill,
1972.
24


Thermal conductivity of comets

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