Experimental study relating thermal conductivity to thermal piercing of rocks

Int. J. Rock Mech. Min. Sci. Vol. 5, pp. 205-218. Pergamon Press 1968. Printed in Great Britain

E X P E R I M E N T A L STUDY R E L A T I N G T H E R M A L C O N D U C T I V I T Y TO T H E R M A L P I E R C I N G

OF ROCKS V. V. MIRKOVICH Mines Branch, Department of Energy, Mines and Resources, Ottawa, Canada (Received 6 September 1967) Abstract--Thermal conductivity of 19 selected Canadian rocks was measured to determine the effect of thermal conductivity on thermal piercing of rocks. Rocks with higher thermal conductivity generally tend to pierce better, but a definite relation could not be established. However, a very good correlation was obtained between the thermal piercing rate and the product of thermal diffusivity and thermal expansion.

1. INTRODUCTION THE subject of thermal conductivity of rocks is of considerable interest in different fields of science and engineering. In devising recovery methods, petroleum technologists have been concerned with thermal conductivity of oil-bearing sandstones and shales, and the effects of oil and tar contents on their thermal properties. Heat transfer engineers have measured thermal properties of rocks and expanded minerals in connection with their use as insulation and heat storage materials. More recently, the diffusion of thermal energy in rocks has become a matter of appreciable interest to scientists investigating the effects of underground nuclear explosions; and thermal conductivity of rocks and minerals has become important in the planning of manned landings on the moon and subsequent lunar exploration. The largest contribution to this field has been made, however, by geologists and geophysicists through their studies of the temperature distribution and heat flow in the earth's crust, of the rate of heat loss from the earth's surface, and of other natural thermal phenomena. Although there is a large amount of published information on thermal conductivity of rocks and minerals, these values are not very useful. Most of the papers deal with specific problems [1-6] and in comparing the data it becomes apparent that there is a relatively wide range of values for what seems to be the same types of rocks. Computation of thermal conductivity, based on the chemical and/or mineralogical composition of rocks does not seem to be feasible at present. The use of published data is not advisable for several reasons: firstly, full descriptions of the rocks has seldom been given; secondly, due to the simplicity of the equipment used, the accuracy of measured conductivities is questionable; thirdly, almost no information has been published on conductivities of rocks at higher temperatures. Attempts to relate thermal conductivity (or diffusivity) to thermal piercing of rocks are not numerous. NORTON [7] developed an equation which relates thermal diffusivity, thermal expansion and maximum strain to 'spalling tendency'. A somewhat similar expression 205

206

V. V. MIRKOVICH

was given by VASILIEV[8], and also by LIDMANand BOBROWSKY[9]. In a more recent publication, FREEMANet aL [10] discuss the mechanism of thermal spalling of rocks in terms of thermal and mineralogical properties of the rocks. However, they do not propose a method for prediction of piercing rates. There have been a number of theoretical and experimental investigations of materials under conditions of thermal stress (CrmNG [11], MAROVELLIet aL [12], GRAY [13]). Although these theoretical mathematical analyses undoubtedly contribute to a better understanding of the problem, from the practical side they contribute very little. For instance, CrlENG [1 I] in his analysis of resistance to thermal shock assumes that the surface heat transfer coefficient, thermal conductivity, modulus of elasticity and Poisson's ratio are constants and that thermal expansion is a linear function of temperature. Similarly MAROVELLIet al. [12] assumed, in their theoretical approach to thermal fragmentation of rock, that thermal and mechanical properties are constant (although, for calculation purposes, these properties had to be evaluated at some average temperature). It is well known that most of the physical properties of rocks vary widely with temperature. A theoretical relationship based on the assumption that these properties are constant is erroneous and will not give realistic predictions of piercing rates. The purpose of the work reported here is, therefore, to measure the thermal conductivity of representative Canadian rock specimens between 100° and 1000°C, and also to establish the relations that exist between thermal conductivity and the tendency of some rocks to spall more than the others. 2. PROCEDURE

2.1 Apparatus and procedure for measurement of thermal conductivity A comparative-method thermal conductivity apparatus was used for this work. Because a detailed description of the equipment appears elsewhere [14], onlyits principal features will be outlined here. The apparatus, schematically shown in Fig. 1, consists of four main components: sample column, heat source, heat sink and heat guard. The sample column is in turn composed of two standards with the unknown sample placed between them, and a heat stabilizer. The heat stabilizer, standards and the unknown sample are cylinders, 25.4 mm in diameter and 25.4 mm high. The sample column is centered within the cylindrical heat guard. It is held between the heat source and the heat sink with the heat stabilizer resting on the heat source.

m IEAT |INK

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STANDARD" i,,~/ SAMPLE~

~GUARD HEATERS

. =

STANDARD-

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f

INSULATION

f-COntAINER ~--THERMOCOUPLES

fEA1 3UR(

k_ FIG. 1. Schematicof thermal conductivityapparatus (not to scale).

STUDY RELATING THERMAL CONDUCTIVITY TO THERMAL PIERCING OF ROCKS

207

By adjusting the power input to the heat source and heat sink, a desired temperature gradient is established in the sample column. Radial heat flow between the column and the heat guard is prevented by matching the temperature of the heat guard to that of the sample column on the same level. For determination of thermal conductivity by comparative methods it is desirable that the conductivity of the standard be similar to the conductivity of the investigated material. Pyroceram Code 9606* seemed to satisfy most of the requirements, and was consequently used for all rock measurements described herein. In principle it does not matter whether the steady state was approached from successively rising or decreasing temperatures. However, as the properties of rocks are permanently altered at high temperatures, and because thermocouples deteriorate faster at high temperatures, the measurements were started at a low temperature. Once the measurement for a given sample had started, it was continued without interruption for some 120-130 hr. 2.2 Materials The rock specimens for this study were chosen from material which was originally obtained for the Jet Piercing Research Project [15]. The 19 samples used in the thermal conductivity measurements were selected and characterized by GELLER and SOLES [16], [17]. Descriptions and properties of the rocks are summarized in Table 1. The piercing ratings are arbitrary numerical values based on the amount of rock displaced per unit time in a jet-piercing test. The approximate ratings are as follows: 10-8 8-6 6-4 0

(Excellent) (Good) (Fair) (Non-piercing)

-----

clean spalls; continuous, rapid piercing evidence of fusion, continuous, slower piercing abundant fusion; slow piercing, intermittent blockage does not spall, calcines or melts.

Linear thermal expansion (GELLER [18]), specific heat (SVIKIS [19]), and density (HANES [20]), were measured on each specimen. 3. RESULTS

Thermal conductivities of 19 samples, representing three groups of rocks, were measured at temperatures ranging from 100 to 900°C. In Figs. 2--4, thermal conductivity of igneous, carbonate and quartzose rocks are plotted against temperature. The curves were fitted to the data by inspection. Some 8-10 conductivity measurements are represented by each curve. The average deviation of the experimental data from the smoothed values is 4- 2.9 ~o. The standard deviation calculated for all data is 4- 0.0009 W/cm°C. To check the reproducibility of the results, thermal conductivity measurements were repeated (over the whole temperature range) on four rocks. New samples were used in each case. The new values differed from those previously determined by an average of + 2 ~o for M-190, --4~o for M-191, +3~o for M-168 and + 3 ~ for M-184. The thermal conductivities of igneous rocks show a fairly wide spread at low temperatures. As shown in Fig. 2, the conductivities at 100°C for this group of rocks vary from 0.019 to 0.029 W/cm°C. With increasing temperature the conductivities decrease, except for sample M-193, which shows a minimum at about 400°C, after which the thermal conductivity increases again. Also, the difference between samples decreases with increasing temperatures: the smallest difference between the highest and the lowest values occurs at approximately 600°C. * Supplied by Coming Glass Works, Coming, New York, U.S.A.

208

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STUDY RELATINGTHERMALCONDUCTIVITYTO THERMALPIERCINGOF ROCKS

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F[o. 4. Thermal conductivity of miscellaneousquartzose rocks. The results for carbonate rocks are shown in Fig. 3. The conductivity of this group of specimens is generally higher and somewhat more temperature dependent (note the difference in the scale between Figs. 2 and 3) than that of the igneous rocks. The conductivities of quartzose rocks are given in Fig. 4. With the exception of M-199, the low-temperature conductivities of these rocks are higher than those of the other two groups. As the temperature increases, however, the conductivities decrease rapidly, approximating the values for the other rocks. 4. DISCUSSION The primary purpose of this study is to determine the significance of thermal conductivity data in the evaluation of thermal piercing properties of rocks. Basically, the relation can be established in two ways: (i) by comparing the piercing ratings (or rates) of rocks and their thermal conductivities, and/or (ii) by comparing the piercing rates and thermal conductivities in conjunction with other physical properties. The curves in Figs. 2-4 reveal an interesting and pertinent feature; the thermal conductivity for most samples diminishes with increasing temperature. Of even greater importance, the substantial difference which exists between the conductivities of different rocks at lower temperatures also diminishes as the temperature increases. The significance of this behaviour will become apparent when its causes are briefly examined. The conductivity of an aggregate, either homogeneous or heterogeneous, depends primarily on the conductivity of the individual crystals and on the thermal contact between these crystals. An exposure to high temperature invariably causes some physical or chemical changes. These changes in turn usually lead to the deterioration of intercrystalline bonds,

210

v.v. MIRKOVICH

and ultimately to the destruction of the rock. Thus, if the conductivity of individual crystals is increased or decreased, the overall conductivity of the rock will change in the same direction. The destruction of the intercrystalline bonds, on the other hand, will always reduce the thermal conductivity owing to the increase of thermal resistance at the grain boundaries, and will ultimately be the overriding factor in establishing the total resistance of the aggregate. Therefore, regardless of the original low temperature conductivity, the high-temperature conductivities of crystalline rocks will be generally low and not much different from each other. Hence, high-temperature conductivities in most cases do not reflect the original nature of the rocks, and as such should not be used in the evaluation of thermal piercing properties of rocks. A satisfactory correlation between the thermal-piercing properties of rocks and their low-temperature conductivities cannot be established from the data in Figs. 2-4. It is apparent that in the case of miscellaneous quartzose rocks, the pierceability increases with

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increasing low-temperature conductivity. Unfortunately the number of measurements is insufficient to serve as a basis for a general conclusion, especially since the conditions in the other two groups are not as orderly. Rock piercing results from high stresses generated by non-uniform expansion owing to temperature differentials formed over relatively short distances within the material. It follows therefore that thermal diffusivity and thermal expansion should be two major factors influencing thermal-piercing of rocks. Although both properties contribute to the formation of the same thermal stresses, their effects are different. Under the same conditions low thermal diffusivity forms steep temperature gradients and consequently high thermal stresses, while low thermal expansion produces low stresses. It is therefore logical to expect that a dividend of these two quantities can be best correlated with the rate of thermal piercing. In Fig. 5 the thermal-piercing rating is plotted against E/a. E is the linear thermal expansion at 300°C, and the thermal diffusivity a = k/pC~, where k is thermal conductivity at 300°C, p is density at 25°C and C, is the average specific heat between 25 ° and 500°C.

STUDY RELATING THERMAL CONDUCTIVITY TO THERMAL PIERCING OF ROCKS

211

The values for density and specific heat are available only at temperatures stated. Fortunately this is not a serious limitation because for most rocks p and Cp change little up to 300--400°C. The numbers appearing at each point identify the rocks. A partial correlation between piercing rating and E/~ can be detected from Fig. 5. It is evident that most of the points fall above the dotted line extended from M-188 to M-193. The minimum piercing rating can thus be established from the ratio of thermal expansion to thermal diffusivity. For instance at E/~ = 45, the minimum piercing rating will be 7. The actual rating could be, however, as high as 10. A marked departure from this correlation is shown by rock M-187, and to some extent by rocks M-184 and M-185. The reason for this deviation must be sought in their chemical compositions. All three specimens are carbonate rocks, which on heating form (thick) oxide layers. Formation of such oxide layers on the surface disrupts the original heat flow to the body of the rock and results in stoppage of the piercing process. SOLES[21] observed that both specimens, M- 184 and M-185, actually spall at the surface, but piercing below the surface is quickly halted, possibly because of calcination. Similarly, the dolomitic limestone M-187 initially spalls rapidly, but then slows down and because of calcination eventually stops. In comparison with M-184 and M-185, the somewhat higher piercing rating of M-187 can be attributed mainly to a higher E/a, but also, perhaps, to the presence of about 25 per cent of dolomite. MgCO3 in dolomite decomposes at a temperature approximately 150°C lower than CaCO~ causing weakening and partial disintegration of the rock before the formation of the insulating oxide layer. The rate of thermal piercing depends on the development of failures in the rock, principally due to the thermal stresses. The failures, however, are governed not only by the maximum stress but also by the distribution of the stress within the body. Thus a steep but shallow temperature gradient, resulting from a combination of a high (surface) heat transfer coefficient and low thermal conductivity will not necessarily cause a failure. In fact, in the case of some rocks an insulating, fused, or oxidized layer is likely to form and disrupt the heat transfer process. A related problem was studied by MANSON[22] who examined, essentially, the effect of different surface heat transfer coefficients on thermal shock resistance of materials with different thermal conductivities. He found that at low surface heat transfer coefficients, alumina was less resistant to thermal shock than beryllia which has considerably higher thermal conductivity. However, at higher heat transfer coefficients he found the reverse was true. Assuming that similar rules can be applied in the case of thermal piercing of rocks, high piercing rates can be expected in rocks with high thermal conductivity, as in the thermal piercing process the heat transfer coefficients from the jet flame to the surface are generally very high. Consequently the product, rather than the dividend, of thermal diffusivity and thermal expansion would correlate better with the rate of thermal piercing. In thermal piercing the heated part of the material tends to expand and create stresses. At a certain sufficiently high temperature, the stresses will equal or exceed the strength of the material and cause it to fail. Therefore, when correlating thermal properties and piercing rates, the values for thermal expansion close to these critical temperatures should be used. In contrast to this, and as already discussed, the high-temperature thermal conductivities or thermal diffusivities are not specific for the majority of rocks. Furthermore, as thermal piercing is a dynamic process, thermal diffusivity at the piercing temperature cannot be a determining factor: the heat contained by a particle at the moment of spalling has been absorbed before the piercing temperature is reached and consequently the heat propagation is governed by lower temperature thermal diffusivity.

212

v.v. MIRKOVICH

The actual temperatures at which thermal diffusivity and thermal expansion should be evaluated is rather difficult to determine theoretically. The critical temperature, for instance, which a portion of the rock must reach to spall, depends not only on a number of physical properties, which are themselves functions of temperature, but also on the heat transfer coefficient between the heat source and the surface of the rock. For this study, therefore, these temperatures were established by trial and error. Figure 6 is a plot of the product of thermal diffusivity and thermal expansion, aE, against the piercing rating p. The values for a were taken at 200°C and for E at 500°C. It is evident that most of the data are within the area bounded by the two dotted lines. The exceptions on the lower side are M-184, M-185, and M-187. As previously discussed, the principal reason for the lower-than-predicted piercing ratings of those rocks is the chemical reaction which takes place at the piercing temperatures. The differential thermal analysis (DTA) of the rocks used in this study supports such a theory. As shown in Fig, 7, the DTA

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for M-184 indicates only one peak. It occurs between 900 ° and 1000°C and corresponds to the decomposition of CaCO3 (the heated surface of M-184 becomes thus coated with an insulating oxide film before the disintegration of the underlying rock can take place). M-185 has a smaller peak preceding the larger carbonate peak. The smaller peak is due to the decomposition of MgCOs; however, the quantity of MgCOa (about 4.5 per cent) is too small to affect significantly the piercing rating. The other extreme is M-186. In this ease the very high pierceability can be attributed to the pressure generated by the expansion of gases within the voids which occupy 19 per cent of the rock. Also, there is an exothermic reaction above 400°C, probably oxidative, which contributes to the weakening of M-186. The remaining three rocks on the high side, M-188, M-190 and M-194, are somewhat problematic. It is possible that SOL~Sand GELr~ER[17] gave them too high piercing ratings. The continuous piercing rate of rocks which do not spall well is usually lower than their initial piercing rate. Since the properties of a rock do not necessarily change with the depth of penetration, the difference in the piercing rates results from the changes outside the rock mass. The two most likely variations are the change in the configuration of the surface on

STUDY RELATINGTHERMALCONDUCTIVITYTO THERMALPIERCINGOF ROCKS

213

which the flame impinges, i.e. from flat to curved, and the change in the heat transfer coefficient from the flame to the surface (due to the change in configuration of that surface). A study of these factors, which is the subject of another investigation*, could disclose the practical steps which are required to achieve the condition where the continuous piercing rate equals or, at least, closely approaches the initial piercing rate. Presently, however, it would be very valuable if the relation between the initial piercing rates and thermal properTemperature,

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Z FIG. 7. Differential thermal analysis of selected rocks. The vertical scale for carbonate rocks M-168, -184, -185, -186 and -187 is reduced by a factor of eight. ties could be established. With such a correlation one could set up an experimental criterion by which continuous piercing rates could be judged. In Fig. 8 therefore, the piercing rate R (cm/sec) is shown as a function of the product aE. As before, the values for ~ were taken at 200°C and that for E at 500°C. The piercing rates were determined by GELLER [23]. On the average, each point represents the mean of several determinations, but for some points (such as M-179, -187, -193 and -209) only one determination was available. Zero piercing rate is attributed to M-184 and M-185, although there is some initial spalling. * In progress; V.V.M.

214

V. V. M1RKOVICH

TABLE 1. CLASSIFICATION, COMPOSITION, TEXTURE, STRUCTURE, PIERCING RATING, LINEAR

SmleI

Mineral composition* ( ~ ) Rock class Quartz

K-Felds

Na-Felds

Ca-Felds

Ferromagnesians and miscellaneous

1. lgneous Rocks M-178

Gabbro (Diabase)

-188 -190 -191

Nepheline Syenite Rhyolite Hbde Syenite

-192

Nordmarkite

-193

Anorthosite

-194 -208

Granite Granodiorite

-209 -210

Granite Qtz Monzonite

12 (Graphm)

46 (Anso)

23 52 73 (Perthite) 69 (Perthite) 1

52 51 17

34 21

31 4

31

33 27

29 20

23 9

8 97 (An~O 51 (AnsO 25 --4

42 (Anso)

24 Aug; 11 Hbde, Blot; 7 Mag. 2 Mag; 23 Nepheline 3 Mag 9 Hbde; 2 Blot ; 3 Mag 7 Hbde; 2 Aug; 1 Mag -~ Hematite 1 Aug; 1 Mag 3 Blot 12 Hbde; 9 Blot; 3 Epidote + Sphene 10 Blot 10 Blot; I Muscovite

2. Miscellaneous Quartzose Rocks M-179 -199 -200 -207

Quartz Hbde-Qtz Gneiss Taconite Quartzite

100 25

(Including Feldspar 7)

f

70 Hbde; 5 Calcite 23 Granerite; 17 Mag 7 Muscovite

3. Carbonate Rocks

M-168 -184 -185 -186 -187

Dolostone (Recryst) Limestone Limestone (Recryst) Dolostone Dolomitic Limestone

Quartz

Calcite

I i

1 1

94 88

J I

72

Dolomite 98 4 10 80 27

Oth s 2 Tremolite 2 Clay; impurities I Diopside; 1 graphite 1 Sulfides; Oxides; 19 Voids 1 Dark impurities

* Hbde: Hornblende, Blot: Biotite, Aug: Augite, Mag: Magnetite. t F: fine-grained (<1 mm), M: medium-grained (1-5 ram), C: coarse-grained (> 5 mm), +*Calculated from true density.

STUDY RELATING THERMAL CONDUCTIVITYTO THERMAL PIERCING OF ROCKS

THERMAL EXPANSION, SPECIFIC HEAT AND DENSITY OF

Texturet

Structure

19 CANADIAN

Piercing rating

215

ROCKS AND MINERALS

~ Therm. Sp. heat exp. to 500°C (W/sec g °C)

Density (g/cms)

M, diabasic-hypidio.

Homogeneous

2

0"45

0"945

2"98

M, hypidiomorphic F, allotriomorphic M, hypidiomorphic

Homogeneous Homogeneous Homogeneous

9 10 4

0"82 0" 74 0"63

0"991 0-991 0"958

2"98 2"67 2"64

C, panidiomorphic

Homogeneous

5

0"68

0"966

2"66

C, hypidiomorphic

Homogeneous

2

0"40

0"979

2"83

M, hypidiomorphic M, hypidiomorphic

Homogeneous Homogeneous

9 6

0"82 0-82

0"996 0-991

2"65 2"77

M, hypidiomorphic C, panidiomorphic

Coarsely gneissic Homogeneous

8 7

0"82 0"85

0"991 1.004

2-66 2"68

Coarse crystals M, metamorphic M, metamorphic F, strained crystals

Homogeneous Banded, gneissic Banded, gneissic Local foliation

10 5 6 9

0"88 0"50 0"70 0-82

1"046 0"966 0"954 1"042

2"66 3"07 3"30 2"67

Coarse crystals F, sedimentary Coarse crystals M, porous

Homogeneous Local banding Homogeneous Large voids

10 0 0 10

0" 92 0"54 0"44 0"84

1"058 1"008 1"008 1"050

2"86 2"75 2"75 2"325

F, fossiliferous

Homogeneous

1"04

1"017

2"75

t

216

v . v . MIRKOVICH

The line R = f(aE) was calculated by the least square method using all piercing rate determinations for each point, with the exception of M-186. (It was stated previously that the extremely high piercing rate of M-186 is considered not to be due to its thermal properties, but to the porosity of the rock and the chemical reactions at higher temperatures.) Hence, the initial piercing rate is: R=-21-3~E--

1.75 × 10 -~

where thermal diffusivity, ct, is in cm2/sec, and linear thermal expansion, E, is expressed as per cent elongation. Excluding M-186, the standard deviation is 0-04 cm/sec. Although the validity of the above relation is likely to be limited to conditions of piercing

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"186

0.5C

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0.0~5

eE

FIG. 8. Relationship between piercing rate and the product aE. a and E were determined at 200°C and 500°C, respectively. with high-temperature jet flames, its usefulness is, nevertheless, considerable as it establishes the pierceability of rocks not only in relative terms, but also predicts their absolute piercing rates. A precaution should be taken, however, to examine the rocks mineralogically and/or to establish what chemical or physical reactions occur at temperatures up to the piercing temperature. For example, the piercing rate of M-179 (Fig. 8) appears to be appreciably higher than predicted. The point M-179 represents only one piercing determination, and it may be possible that under the particular conditions of that test, too high a piercing rate was obtained. On the other hand, knowing that this is a quartz rock which at 573°C undergoes a volume changing crystal phase transition, a higher-than-predicted piercing rate could be expected. Furthermore, since its fusion temperature ( > 1700°C) is much higher than that of the other rocks, considerably greater temperature gradients can be induced and thus higher piercing rates attained.

STUDY RELATINGTHERMALCONDUCTIVITYTO THERMALPIERCING OF ROCKS

217

In a purely theoretical derivation which relates thermal properties to spalling, four fundamental properties must be considered: thermal conductivity (or diffusivity), thermal expansion, strength, and elasticity modulus of the material. This discussion would be incomplete if no mention were made of the significant fact that two of these factors, the strength of the material and its modulus of elasticity, were not employed in this analysis. Nevertheless, a very good correlation between pierceability and thermal properties alone was obtained. An entirely satisfactory explanation cannot be offered at this stage. Possibly the strength and elasticity of rocks, properties which are usually measured at room temperature, are not relevant at piercing temperatures, because, as with thermal conductivity, they may not vary significantly from one sample to another. It should also be added in conjunction with this that some thermal properties, such as thermal expansion, frequently vary with the temperature in such a way that they cannot be described by a continuous function. This is probably one of the severest limitations to any theoretical derivation relating pierce ability to thermal properties because one is compelled to introduce crippling simplifications by assuming that most of the thermal properties are constants or, at best, simple functions of the temperature. In concluding this discussion, it should be noted that the principal limitation of the experimental correlations obtained in this work is that the thermal piercing of rocks does not depend solely on the properties of rocks. The effect of the rate of heat transfer from the heating medium to the surface, and the effect of configuration of the heated surface on temperature distribution in the rock should be considered in any experimental or theoretical analysis of the rock piercing process. 5. CONCLUSIONS On the basis of the experimental evidence of this investigation, the following conclusions were drawn. (i) Thermal conductivity of rocks decreases with increasing temperature. The difference between the thermal conductivities of individual rocks diminishes with increasing temperature. High temperature conductivities generally do not reflect the original nature of the rocks. (ii) Rocks with higher conductivity generally pierce better. However, a simple relation between thermal conductivity and thermal piercing of rocks cannot be established. (iii) Good correlation was obtained between the piercing rating (and/or piercing rate) and the product of thermal diffusivity and thermal expansion. It is significant that only the product of thermal diffusivity and thermal expansion was sufficient for this correlation. In any theoretical derivation four basic properties are normally considered: thermal conductivity (or diffusivity), thermal expansion, strength and modulus of elasticity. (iv) For the correlation obtained in this work, thermal conductivity and thermal expansion must be evaluated at different temperatures because with increasing temperature, their values vary in the opposite directions. (v) The principal limitation of the correlation is due to 'external' factors. Thermal piercing depends not only on the properties of the rock, but also on the rate of heat transfer to the surface of the rock and on the configuration of that surface. Effects of these two 'external' factors on the piercing rate should be investigated. Acknowledgements The author acknowledgeshis indebtedness to Dr. J. A. SOLESand Mr. R. M. BUCHANAN of the Mineral Processing Division for their generous contribution in discussing the mineralogical and other aspects of this study.

218

V. V. MIRKOVICH

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