Effective Thermal Conductivity of Fluid-Saturated ... - ACS Publications

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Ind. Eng. Chem. Res. 2002, 41, 3586-3593

Effective Thermal Conductivity of Fluid-Saturated Porous Mica Ceramics at High Temperatures and High Pressures I. M. Abdulagatov,* S. N. Emirov, Kh. A. Gairbekov, M. A. Magomaeva, S. Ya. Askerov, and E. N. Ramazanova Institute for Geothermal Problems of the Dagestan Scientific Center of the Russian Academy of Sciences, 367003 Makhachkala, Shamilya Street 39-A, Dagestan, Russia

The effective thermal conductivity (ETC) of fluid-saturated porous mica ceramics with open pores was measured over a temperature range from 275 to 423 K and at pressures up to 400 MPa using a steady-state parallel-plate apparatus. It is an absolute, steady-state measurement device with an operational temperature range of 273-1273 K and a pressure range up to 1500 MPa. We used argon and water as pore saturants. The estimated accuracy of the method is (2%. The porosity of the samples was 2%, 14%, and 26%. The effect of pressure, temperature, and porosity on the ETC behavior of the fluid (Ar and H2O)-saturated porous mica ceramic was studied. A sharp increase of the ETC was found for porous mica ceramic with gas (Ar) saturated at low pressures (between 0.1 and 100 MPa) along the various isotherms, while for the same sample saturated with water, the pressure dependence of the ETC displayed very weakly. The measured values of the ETC for a fluid-saturated porous mica ceramic were compared with the values predicted by various theoretical and semiempirical models. The effect of the size, shape, and distribution of the pores on the ETC of porous mica ceramic was discussed. 1. Introduction The problem of determining of the effective thermal conductivity (ETC) of porous media is of interest to a wide range of engineers and scientists. The thermal method of oil recovery and shale oil retorting operations represent problems for which knowledge of the ETC of fluid-saturated porous media at high temperatures and high pressures is essential. Heat conduction in porous materials plays an important role also in energy transport for a number of practical and technical processes. In thermal energy storage devices, artificial heating and cooling of buildings, weather control, thermal exchange in heat pump systems, geothermal operation and drying of food grains, space technology, aviation, high-temperature furnaces, and metallurgy, the ETC values of porous materials are needed. The thermal conductivity measurements are very important also for studying of the heat transport phenomenon mechanisms in various solid-state structures. The ETC values of porous materials are affected by various factors, among which the most important are temperature, pressure, porosity, and microstructure. The microstructure (sizes and shapes) and distribution of the pores significantly effect the heat-transfer processes in porous media. The thermodynamic state of the fluid phase present in the pores plays also important role among the many factors which influence thermal conductivity. Determination of the thermal conductivity for fluid-saturated porous media is a difficult problem because of the coupled nature of heat-transfer phenomena. * To whom correspondence should be addressed. E-mail: [email protected]. Fax: (8722) 67-20-67. Tel: (8722) 6266-23. Present address: Physical and Chemical Properties Division, National Institute of Standards and Technology, 325 Broadway,Boulder,CO80303.E-mail: [email protected]. Fax: (303) 497-5224. Tel: (303) 497-4027.

Only limited experimental ETC data for porous materials under pressure are avaliable in the literature.1-5 Basically, the measurements of the ETC of porous materials reported in the literature were performed at high pressures but at low temperatures. Hughes and Sawin1 have made measurements of the thermal conductivity of several insulators at high pressures (up to 1200 MPa) and at temperatures between 273 and 450 K with uncertainties within (5-6%. They found a rapid increase of the thermal conductivity by nearly a factor of 2 in the range between 0 and 6 kbar. Horai and Susaki3 have made measurements of the thermal conductivity of silicate rock at temperatures from 300 to 700 K and at pressures up to 12 kbar with an accuracy of (4-5% using a steady-state method. The thermal conductivities of rocks at pressures up to 5 kbar and at room temperature have been reported by Seipold et al.4 using a flash method. In works in refs 6-30 were derived expressions for the prediction of the ETC of multiphase (two- and threephase systems) porous materials. The purposes of this study are to provide accurate experimental ETC data for porous material (mica ceramic) saturated with different fluids (argon and water) and with different porosities at temperatures from 275 to 423 K and at pressures up to 400 MPa, to study the effect of temperature, pressure, and porosity on the ETC behavior of fluid (argon and water)-saturated porous mica ceramics, and to test the validity of the various theoretical and semiempirical expressions for the prediction of the ETC of fluid-saturated porous systems. To measure the thermal conductivity of dry solids and rocks, which are nonhomogeneous, a parallel-plate method was used in our previous works.31-34 In this work we have measured the ETC of fluid (argon and water)-saturated mica ceramic porous samples with an estimated accuracy of (2% for porosities of 2%, 14%, and 26% at temperatures

10.1021/ie0200196 CCC: $22.00 © 2002 American Chemical Society Published on Web 06/19/2002

Ind. Eng. Chem. Res., Vol. 41, No. 15, 2002 3587

from the heater to the upper and lower specimens; Q1 ) λS1/h1∆T1 and Q2 ) λS2/h2∆T2 are the heat flows transferred by conduction through the lower and upper specimens, respectively; Qlos is the heat losses through the lateral surface of the samples; S1 and S2 are the cross-sectional areas of the specimens that heat flows through; h1 and h2 are the heights of the samples; and ∆T1 and ∆T2 are the temperature differences across the samples thickness. The thermal conductivity is obtained from the measured quantities Q, Qlos, ∆T1, ∆T2, S1, S2, h1, and h2. The heat flow Q from the heater is distributed between the two studied samples Q1 and Q2. The values of Q were corrected by a specimen side loss factor Qlos. The values of Qlos can be estimated from the relation

2πh Qlos ) λm∆T ln(d/D) Figure 1. Schematic representation of the apparatus for highpressure and high-temperature thermal conductivity measurements of porous materials: (1 and 2) samples; (3) heater; (4 and 5) coolers; (6) spring; (7) guard heater; (8) sample for electrical property measurements; (T1, T2, T3, T4, and G) thermocouples.

from 275 to 423 K and at pressures up to 400 MPa using a steady-state parallel-plate technique. 2. Experimental Procedure The experimental arrangement used for the present measurements is similar to that described in our previous several publications.31-34 Because the details of the apparatus, the construction of the thermal conductivity cell, and the experimental procedure have been described in previous publications,31-34 they will only be briefly reviewed here. The experimental apparatus consists of a high-pressure chamber, a thermal conductivity cell, an air thermostat, a high-precision temperature regulator, and high-pressure liquid and gas compressors. Figure 1 shows a schematic drawing of the thermal conductivity cell. In this method, thermal conductivity is obtained from simultaneous measurements of the steady-state heat flux and temperature gradient in the sample placed between the heating and cooling plates. Two thermocouples (T2 and T3) were embedded in the center of the inner surface of the bronze disk. The heater is located between these thermocouples. The other two thermocouples (T1 and T4) were soldered to the body of the heater 3. The temperature difference and temperature of the chamber were measured with copper-constantan thermocouples (T1, T2, T3, and T4). The pressure was created with liquid and gas compressors (Unipress Type GCA, Poland). The pressure in the chamber has been measured with a manganin manometer with an uncertainty of 0.25%. The high-pressure chamber is located in the air thermostat. The temperature in the air thermostat was controlled automatically to within (5 mK. The thermal conductivity λ of the specimen is deduced from the relation

λ)

Q - Qlos S2 S1 ∆T1 + ∆T2 h1 h2

(1)

where Q ) Q1 + Q2 + Qlos is the heat flow transferred

(2)

where d ) 12 mm and D ) 22 mm are the diameter of the sample and the inner diameter of the high-pressure chamber, respectively; h ) h1 + h2 is the height of the samples; λm is the thermal conductivity of the media of transmitted pressure (oil); and ∆T ) Tm - TC is the temperature difference between average values Tm ) (T2 + T1)/2 or Tm ) (T3 + T4)/2 of the temperatures T2 and T3 of the lateral surface of the specimens and temperatures T1 and T4 of the inner surface of the highpressure chamber; TC ) T1 ) T4. The values of the temperature difference are almost constant, ∆T ≈ 1.2 K. The heat losses by conduction along the electrical leads, by radiation and heating, are negligibly small.31,32 The uncertainties of all measured quantities are δQ ) 0.57%; δQlos ) 2%; Q ) 0.28 W; Qlos ) 0.02 W; ∆T1 ) 2 K; ∆T2 ) 1.5 K; δS1,2 ) 0.33%; δh1,2 ) 0.33%; and δ(∆T1,2) ) 0.1%. The propagation of uncertainties related to the uncertainties of pressure, temperature, and height are 0.03%, 0.01%, and 0.002%, respectively. Heat losses through the side surface of the specimens were 3.5% of the total amount of heat supplied to the specimens. The total uncertainty in the thermal conductivity measurement stems from uncertainties in measured quantities of not more than (2.0%. To check the reproducibility, the measurements at each experimental temperature T and pressure P were repeated 5-10 times. The scatter of the experimental results did not exceed (0.5%. The measurements were made with temperature differences ∆T1 between 1.5 and 2 K. In porous materials, heat is propagated basically by thermal conductivity through the solid, by radiation, and by convection through the pores. When the sizes of the pores are small (r ≈ 10-6 m; therefore a small temperature difference across the pores, ∆Tpore ≈ 0.003 K), convection can be neglected. This makes it possible to minimize the risk of the convection in the pores. The absence of convection in the pores was verified experimentally by measuring the thermal conductivity with various temperature differences ∆T1. Heat transfer by radiation increases as the pore size is increased, and its effect can be calculated by the method described in refs 31 and 32. Because it has a T 3 (λrad ) 4×a6σT 3r) dependence, variation obviously plays an increasingly active role at high temperatures. In this work it has been assumed that the solid phase is transparent to thermal radiation. Therefore, heat transfer through the pores by radiation can be neglected. Table 1 shows the characteristics of the samples with their porosities, pore and sample sizes, densities, and

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Ind. Eng. Chem. Res., Vol. 41, No. 15, 2002

Table 1. Characteristics of the Samples sound diameter, height, density velocity -3 -3 -3 sample 10 (m) 10 (m) (kg‚m ) (km‚s-1) 1 2 3

12 12 12

3-3.5 3-3.5 3-3.5

2103 2218 2690

2.570 3.101 4.110

pore size (µm)

open porosity (%)

1.0-3.5 0.8-3.0

26 14 2

Table 2. Experimental Values of the ETC of Gas (Ar)-Saturated Porous Mica Ceramics for Various Temperatures and Pressures

Table 3. Experimental Values of the ETC of Liquid (H2O)-Saturated Porous Mica Ceramics for Various Temperatures and Pressures ETC, λeff (W‚m-1‚K-1), at pressures P (MPa) T (K)

0.1

20

275 323 373 423

1.18 1.39 1.59 1.82

100

150

200

250

1.22 1.43 1.64 1.86

m ) 26% 1.25 1.29 1.47 1.52 1.68 1.72 1.90 1.95

1.31 1.53 1.74 1.97

1.32 1.54 1.75 1.98

1.33 1.54 1.76 2.01

275 323 373 423

1.66 1.70 1.73 1.77

1.70 1.74 1.78 1.83

m ) 14% 1.76 1.82 1.80 1.87 1.85 1.91 1.87 1.94

1.85 1.91 1.95 1.97

1.87 1.92 1.95 2.01

1.88 1.93 1.97 2.02

ETC, λeff (W‚m-1‚K-1), at pressures P (MPa) T (K)

0.1

20

50

275 323 373 423

0.62 0.75 0.86 0.96

1.07 1.16 1.28 1.35

275 323 373 423

1.06 1.10 1.13 1.17

275 323 373 423

2.42 2.51 2.59 2.68

100

150

200

250

300

350

400

1.24 1.36 1.43 1.51

m ) 26% 1.37 1.43 1.45 1.46 1.52 1.54 1.54 1.59 1.61 1.61 1.67 1.70

1.47 1.55 1.62 1.71

1.48 1.56 1.63 1.73

1.49 1.57 1.64 1.74

1.50 1.58 1.66 1.75

1.26 1.31 1.34 1.38

1.43 1.45 1.49 1.53

m ) 14% 1.56 1.64 1.68 1.59 1.68 1.72 1.62 1.71 1.76 1.65 1.74 1.80

1.72 1.76 1.81 1.84

1.75 1.80 1.84 1.87

1.76 1.80 1.84 1.88

1.77 1.81 1.85 1.88

2.95 3.04 3.11 3.18

3.36 3.43 3.50 3.56

m ) 2% 3.48 3.49 3.50 3.55 3.56 3.57 3.61 3.62 3.63 3.66 3.68 3.69

3.51 3.58 3.64 3.70

3.52 3.59 3.65 3.71

3.53 3.60 3.66 3.72

3.54 3.61 3.67 3.73

sound velocities. Sound velocity is one of the important characteristics of the porous solids because there is a good correlation between the sound velocity, density, and thermal conductivity3 of the porous solids. The structure of the samples was analyzed using a scanning electron microscope. The porous mica ceramic was prepared in a normal press by compressing (at pressures of 1 kbar and at temperatures of 500 °C) a mixture of muskovite [40%; KAl2(OH3)2(AlSi3O10)] and ceramic (60%; caoline; Al2O3‚2SiO2‚2H2O) powders (having an average particle size of 0.1-3 µm) into pellets of the required dimensions in air media. The pores have a cylindrical shape. Before measurements, the samples were dried at a temperature of 120 °C for 5-6 h and then were slowly cooled. The studied specimens were cylindrical in shape with a 3-3.5 mm height and a 12 mm diameter. Porous mica ceramic samples contained chaotic and uniformly distributed, open, and interconnected pores with random orientation. Hydrostatic pressure was applied to the samples. The sample was completely saturated (filled) with fluid using a special method developed in our previous works.31,32,34 Fluid (argon and water)-saturated mica ceramic porous materials with porosities of 2%, 14%, and 26% were used.

50

Figure 2. ETC of argon- and water-saturated porous mica ceramic samples as a function of pressure: (1) water-saturated sample; (2) argon-saturated sample.

3. Results and Discussion

Figure 3. Experimental ETC of argon-saturated porous mica ceramic samples as a function of the temperature for various porosities. The solid curves are guides to the eye.

Results for 176 measurements of the ETC of fluidsaturated porous mica ceramic at various porosities and at temperatures from 275 to 423 K and at pressures up to 400 MPa are reported in Tables 2 and 3. Measurements were made for two fluid (argon and water)saturated (completely filled with water and argon) samples with porosities of 2%, 14%, and 26%. Figures 2-4 show the experimental ETC of the water- and argon-saturated mica ceramic samples with various porosities as a function of pressure, temperature, and porosity. A sharp increase of the ETC was noted at low pressures (below 100 MPa), and it leveled off at high pressures for gas (argon)-saturated porous samples,

while for liquid (water)-saturated samples the pressure dependency of the ETC displayed very weakly (see Figure 2). Most thermophysical, electrical, and acoustical properties of porous materials show a typical pressure dependence.35-40 In the low-pressure range, a rapid rise of the thermophysical properties of porous materials with increasing pressure P is observed. Then, at high pressures the ETC λeff increases almost linearly with pressure P. These are all general properties of the porous systems. The increased slope in the initial pressure range is generally explained by the closing of pores (some pores close completely and others become narrow) and cracks and increasing mechanical contacts

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Figure 4. Experimental ETC of argon-saturated porous mica ceramic samples as a function of the porosity for various temperatures. The solid curves are guides to the eye.

Figure 6. Porosity of the argon-saturated mica ceramic sample as a function of the pressure derived from experimental ETC data along various isotherms. Table 4. Parameters C1 and C2 for Equation 3 T (K)

Figure 5. Schematic representation of the pressure behavior of porosity m and volume compressibility βT for porous materials.

between the grains.41 The typical crossover pressure for porous mica ceramic samples is observed at pressures of about 75-100 MPa. The crossover pressure results from the sharp change in volume compressibility βT ) (1/V0) (dV/dP) of porous media with increasing pressure.35,42 Figure 5 shows schematically the typical pressure behavior of porosity m and compressibility βT of the porous materials. The part of the compressibilitypressure curve (βT-P) or porosity-pressure curve (mP) (see Figure 5) above 100 MPa varies slightly linearly with pressure P. This can be explained as a result of changes of the fraction of pore volumes (a decrease in the porosity)35-38,40 or an increase of the density of the sample. For example, at pressures of about 400 MPa the porosity of the sample changed about 1-2%. Therefore, the ETC change due to the porosity change is about 0.04-0.07 W‚m-1‚K-1. Figure 6 shows the pressure dependence of the porosity of the argon-saturated mica ceramic derived from the present ETC measurements for different temperatures. Because of the compressibility of the liquids (water) being much lower than that of the gases (argon), the pressure effect on the ETC of a water-saturated porous sample is smaller than that of an argon-saturated sample (see Figure 2). At pressures above 100 MPa, the thermal conductivity is a weak linear function of pressure. The ETC of fluidsaturated porous mica ceramic increases monotonically (almost linearly) as the temperature increases along each measured isobar (see Figure 3; for example, for argon-saturated mica ceramic along the isobar of 400

C1

C2

χ2

275 323 373 423

Ar-saturated mica ceramic (m ) 26%, T0 ) 273 K, P0 ) 22 MPa) 845.55 20.38 967.94 19.33 1093.37 18.23 1238.48 18.22

1.23 1.32 1.12 1.26

275 323 373 423

H2O-saturated mica ceramic (m ) 26%, T0 ) 273 K, P0 ) 113 MPa) 767.60 24.50 968.09 25.72 1189.95 27.46 1451.59 30.03

1.09 0.97 1.18 1.35

MPa) for both water- and argon-saturated porous specimens. However, the temperature coefficients of the ETC, (1/λeff) (∂λeff/∂Τ)P, for water-saturated samples depend more strongly on the porosity than those for argonsaturated specimens do, while the pressure dependency of the temperature coefficients of the ETC is very weak. The water-saturated porous specimens show much higher temperature coefficients than the argon-saturated samples. This is the result of the effect of the higher thermal conductivity of liquids (water) than of gases (argon). Figure 4 shows the ETC of fluid-saturated porous mica ceramic specimens as a function of porosity m at fixed temperatures and atmospheric pressure (0.1 MPa). The ETC data for specimens with porosities of 26% saturated argon and water were fitted to the expression proposed by Hughes and Sawin1

λeff )

C1 C2 T + T0 P + P0

(3)

The values of parameters C1 and C2 for gas- and watersaturated mica ceramic specimens with porosities of 26% at fixed temperature 423 K are given in Table 4. Singh et al.20 and Pande et al.43 reported an expression for the calculation of the ETC of porous materials at normal and different interstitial air pressures. The final form of the equation is

P λeff ) λn P + P0 where P0 is the characteristic pressure and λn is the ETC

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at normal pressure. This expression can be used to calculate the ETC of porous material at high pressures by means a slightly modified equation such as

Table 5. Reference Thermal Conductivity Data for Water λH2O, Argon λAr, and Skeleton λsol (Solid Material) at Atmospheric Pressure for Various Temperatures T (K)

P λeff ) λn + λ0 P + P0

(W‚m-1‚K-1)43

(3f - 1)λs + (3m - 1)λflu + 4 (3f - 1)λs + (3m - 1)λflu 4

x(

)

λeff )1λs

2

+

m 1 1-m 1 - λflu/λs 3

(

m λs 1-m + 2 λflu - λs

)

λH2O λAr (W‚m-1‚K-1)43 λsol (W‚m-1‚K-1), this work

373

423

0.5646 0.0165 2.4311

0.6434 0.0188 2.5100

0.0251 0.0212 2.6013

0.0288 0.0234 2.6902

pores, m is the porosity, and λeff is the ETC of fluidsaturated porous media. The thermal conductivity of argon is negligibly small (λs >> λflu; see Table 5) compared with that of the solid; therefore, eq 6 for gassaturated porous media can be rewritten in a simpler form as

1-m λeff ) λs 1+m

(5)

(6)

where λs is the thermal conductivity of the skeleton of porous media (solid material, m f 0; in our case λs is the thermal conductivity of muskavite), λflu is the thermal conductivity of a fluid (argon or water) in the

(7)

The thermal conductivity of water in the temperature range from 275 to 423 K and at pressures up to 400 MPa varied between 0.564 and 0.915 W‚m-1‚K-1.44 Therefore, the thermal conductivity of water was still much smaller that λs (2.43-2.69 W‚m-1‚K-1). The values of the thermal conductivity of argon and water are functions of temperature and pressure. The reference thermal conductivity data for water and argon developed at NIST were used for all of the calculations. Temperature and pressure dependences of λflu for argon and water were calculated from Reference Database program REFPRO.44 For the lack of the temperature dependence data for the thermal conductivity of the skeleton (muskavite), the values of λs for various temperatures were calculated by extrapolation of the porosity dependence of the ETC to m f 0 (see Figure 4). The results are given in Table 5. This table contains also the values of the thermal conductivity of argon and water as a function of temperature at atmospheric pressure calculated from REFPRO.44 Missenard18 proposed equations for the calculation of the ETC of various porous materials

λsλflu (4) 2

where f ) 1 - m. The expression (4) was derived for the multiphase media having nonstretched particles. He assumed that all of the phases are spherical in shape and have the same size. For fluid-saturated porous systems with thermal conductivity of the solid material and a saturated fluid ratio (λs/λflu) between 30 and 100, the most appropriate thermal conductivity model is proposed by Odalevskii6 for open-pore media. This model yields the following expression for the ETC:

λeff ) λs 1 +

323

(3a)

where λ0 is the value of the ETC at low pressures (P f 0) and λn + λ0 is the value of the ETC at high pressures (P f ∞). The results of the calculation of the ETC for argon-saturated porous mica ceramic specimens from eq 3a (with λn ) 0.828 W‚m-1‚K-1 and λ0 ) 0.959 W‚m-1‚K-1) for a porosity of 26% are given in Figure 2. Both eqs 3 and 3a represent our experimental ETC data with the same accuracy (see Figure 2). Because of the irregularity of the microstructures, theoretical calculation of the ETC of porous materials, especially for fluid-saturated porous materials, is rather difficult and sometimes impossible. Existing prediction methods are based on certain simplifications such as parallel cylinders, spheres dispersed in a conducting medium, etc. Even with a well-defined microstructure, the problem remains complex because of the existence of the interface resistance. A semiempirical approach is the only practical way of predicting the ETC of porous materials. Therefore, the models for calculation of the ETC λeff strongly depend on real material’s structure and microgeometry of dispersion. A large number of theoretical and semiempirical models6-30 have been developed for the prediction of the ETC of multiphase porous materials. An extensive review of the literature on the ETC of fluid-saturated porous materials was performed by Odalevskii.6 A variety of approaches have been developed by Odalevskii6 to estimate the ETC of a two-phase porous system. The following prediction equations for the calculation of the ETC of fluidsaturated porous materials were proposed by Odalevskii6

λeff )

275

(

λeff ) mλflu + (1 - m)λs

(8)

λeff ) 2mλ1 + (1 - 2m)λ2

(9)

where λ1 and λ2 are defined as

1-

(

λflu λs

(

)

)

λ1 ) λflu 1 + m λflu 1 - m1/3 1 λs

1-

(

λs λflu

)

)

λ2 ) λ s 1 + m λs 1 - m1/3 1 λflu

Equation 8 is valid only for the low porosities. Equation 9 was derived for the porous materials filled with gas. Mendel17 developed an equation to calculate of the ETC of fluid-saturated porous materials

(

λeff ) λs 1 -

)

3m(λs - λflu) 2λs + λflu

(10)

Ind. Eng. Chem. Res., Vol. 41, No. 15, 2002 3591 Table 6. Comparison between Experimental and Calculated Values of the ETC for Water- and Argon-Saturated Porous Mica Ceramics for Various Porosities m ) 14% at pressure 0.1 MPa (water saturated)

m ) 26% at pressure 0.1 MPa (water saturated)

T (K)

λeff(eq 6) (W‚m-1‚K-1)

λeff(eq 14) (W‚m-1‚K-1)

λeff(exp) (W‚m-1‚K-1)

λeff(eq 6) (W‚m-1‚K-1)

λeff(eq 14) (W‚m-1‚K-1)

λeff(exp) (W‚m-1‚K-1)

275 323 373 423

2.04 2.12 1.97 2.04

1.66 1.78

1.66 1.70 1.73 1.77

1.62 1.65 1.31 1.64

1.30 1.43

1.18 1.39 1.59 1.82

sample

λ, eq 4

λexp

λ, eq 5

λ, eq 8

argon saturated m ) 2% argon saturated m ) 14% water saturated m ) 14% water saturated m ) 26%

2.42 1.06 1.66 1.18

T ) 275 K and P ) 0.1 MPa 2.36 2.36 2.38 1.92 1.98 2.09 2.08 2.09 2.17 1.80 1.83 1.94

argon saturated m ) 2% argon saturated m ) 14% water saturated m ) 14% water saturated m ) 26%

2.51 1.10 1.70 1.39

T ) 323 K and P ) 0.1 MPa 2.44 2.44 2.46 1.99 2.02 2.16 2.17 2.18 2.25 1.88 1.92 2.02

porosity (%)

λ, eq 15

λ, eq 9

λ, eq 12

2.35 1.92 2.08 1.78

2.21 1.09 1.50 1.46

2.26 1.45 2.09 1.81

2.44 1.99 2.16 1.86

2.34 1.50 2.17 1.90

λexp

λ, eq 13

λ, eq 11

λ, eq 16

26 14 2

0.62 1.06 2.42

T ) 275 K and P ) 0.1 MPa (Argon Saturated) 1.57 1.57 1.93 1.92 2.55 2.33 2.33

1.59 1.96 2.36

0.90 1.42 2.15

26 14 2

0.75 1.10 2.52


1.65 2.02 2.45

0.93 1.46 2.22

26 14 2

0.86 1.13 2.59


1.71 2.09 2.52

0.96 1.51 2.39

26 14 2

0.96 1.17 2.68


1.77 2.17 2.61

0.99 1.57 2.38

porosity (%)

λ, eq 11a

λ, eq 10

λ, eq 11

λexp

λ, eq 16

26 14

T ) 275 K and P ) 0.1 MPa (Water Saturated) 1.39 1.91 1.66 2.09

0.93 1.42

26 14


0.93 1.47

26 14


0.96 1.52

26 14


1.00 1.57

The expressions (5) and (10) have been derived for the heterogeneous systems with a disperse phase having the shape of a parallel cubic. One of the earliest models was that of Maxwell,22 who developed an equation for the calculation of the ETC for randomly sized spheres of one medium randomly distributed in another medium. The ETC of such porous systems is given by

λeff ) λs

1 + 2χ - 2m(χ - 1) 1 + 2χ + m(χ - 1)

λeff ) λs(1 - 1.5m) Bruggeman19 succeeded in generalizing Maxwell’s spherical pore result to high porosities as

λeff ) λs(1-m)(1+km)λflum[1-k(1-m)]

(11)

(12)

where

where χ is the ratio of the thermal conductivity of the continuous and dispersed phases λs/λflu. If it is assumed that the thermal conductivity of the fluid is negligible compared with that of the solid, eq 11 becomes

1-m λeff ) λs 1 + 0.5m

For low values of porosity m, this equation can be written as

(11a)

k)

λs - λflu 3 2 (2 λ + λ )( λ + 2 λ ) x s x flu x s x flu

For a gas-saturated sample at temperatures 275 and 323 K, the values of k are 0.305 and 0.295, respectively.

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Table 7. Effect of Saturated Fluid in a Mica Ceramic with Various Porosities eq 17 minimum (λs - λeff)/λs

(λs - λeff)/λs

eq 18 maximum

experiment in this work

Water Saturated (λflu/λs) ) 0.243 m ) 2% 0.02 0.03 m ) 14% 0.13 0.19 m ) 26% 0.24 0.32

0.31 0.43

Water Saturated (λflu/λs) ) 0.008 m ) 2% 0.03 0.47 m ) 14% 0.19 0.87 m ) 26% 0.34 0.93

0.02 0.55 0.70

Equation 12 is valid for the heterogeneous systems which consist of stretched ellipsoids and spherical particles. Brailsford and Major23 have extended the results of the Maxwell model22 to cover the full range of porosity, by regarding a random two-phase assembly composed of the two single phases in the correct proportions, embedded in a random mixture of the same two phases having a conductivity equal to the average value of the conductivity of the two-phase assembly. This leads to a value for the ETC of the assembly given by

1 λeff ) λs[(3m - 1)χ + 2 - 3m + {[3m - 1)χ + 2 4 3m]2 + 8χ}1/2 (13) Ziman7 proposed the model which yields the relation between the ETC and porosity for fluid-saturated porous materials

λeff )

λsλflu λsm + λflu(1 - m)

(14)

Sugawara and Yoshizawa45 used an empirical approach to calculate the ETC of wet sandstone

λeff ) (1 - K)λs + Kλflu

(15)

where K ) 2n[1 - (1 + m)-n]/(2n - 1). The optimal value of n for fluid-saturated mica ceramic samples is 0.2. Assuming a regular geometry of the dispersed phase, an integrated theory for the ETC of all kinds of twophase materials was developed by Pande et al.21 For the case λflu/λs f 0 (the thermal conductivity of a saturated fluid phase is too low), the expression for the ETC is given as

λeff ) λs(1 - 1.545Ψ2/3)

temperatures, the deviations are reached up to 12%. For low porosities, the agreement between the values calculated from eqs 11, 11a, and 15 is excellent (3%), while at high porosities, the deviations reached up to 50% and more. Walsh and Decker35 proposed the following equations to estimate the maximum and minimum possible values of the ETC λeff of fluid-saturated porous materials with low porosity (m

Effective Thermal Conductivity of Fluid-Saturated ... - ACS Publications

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