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Damage of basalt induced by microwave irradiation
Minerals Engineering 31 (2012) 82–89
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Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
Damage of basalt induced by microwave irradiation P. Hartlieb a,⇑, M. Leindl b, F. Kuchar c, T. Antretter b, P. Moser a a
Lehrstuhl für Bergbaukunde, Bergtechnik und Bergwirtschaft, Montanuniversität Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria Institut für Mechanik, Montanuniversität Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria c Institut für Physik, Montanuniversität Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria b
a r t i c l e
i n f o
Article history: Available online 16 February 2012 Keywords: Microwave irradiation Finite elements Rock heating and damage
a b s t r a c t In this work microwave irradiation on cylindrical samples of basaltic rock is investigated by laboratory experiments and compared with results from numerical models. Due to the temperature gradient in the samples induced by the microwave irradiation a significant damage indicated by a reduction of the sound velocity and finally the formation of cracks occurs. Applying a microwave power of 3.2 kW leads to a surface temperature of 250 °C and to 400 °C in the centre of a cylindrical sample after 60 s of irradiation. Temperature rise goes along with the formation of both axial and radial cracks. Cracks are not bound to the mineralogical composition but their development is governed by macroscopic temperature gradients and the geometry of the sample. A thermal and a thermomechanical finite element model are formulated and used to calculate temperature distributions and induced thermal stresses. The results indicate that tensile stresses exceed tensile strength leading to cracks as observed experimentally. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Weakening rocks by means of microwave irradiation has been a major topic in research for the past 50 years. The focus of all the projects lies in the development of more economic technologies for breaking, cutting and comminution of various types of rocks. A detailed review of all the work done before the 1980s is provided by Santamarina (1989). Recent activities mainly focus on comminution of various rock types. They have been performed by Walkiewicz et al. (1988), Satish et al. (2006), Jones et al. (2005) and many more. Pickles (2009) gives a detailed overview possible applications in extractive metallurgy. The absorption of microwave energy of a sample depends on high-frequency dielectric properties of the constituents of a rock. These can be described by = 0 + i00 = 0(j0 + ij00 ). Where j0 is the real part, j00 the imaginary part of the relative dielectric constant and 0 is the permittivity of free space. According to Santamarina (1989) typical values for rocks range from 103–50 (j00 ) and 2–10 (j0 ). Values for j00 of basalt range from 0.08 to 0.8, whereas values for j0 vary between 5.4 and 9.4. Several papers are devoted to explaining rock breakage by the different microwave absorption rates and thus different heating rates of the mineralogical constituents of a sample. Whittles et al. (2003), Jones et al. (2005) postulate that inducing fractures between an ore and the host rock can be achieved by using the differential absorption of microwave energy between the different ⇑ Corresponding author. E-mail address: [email protected] (P. Hartlieb). 0892-6875/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mineng.2012.01.011
mineral phases. Together with Jones et al. (2007), Ali and Bradshaw (2009) it is demonstrated that damage after microwave-treatment is caused by tensile stresses occurring during the thermal expansion of the absorbing phases as well as shear stresses along the grain boundaries. It is also shown that particle size as well as delivery method and power density play a major role in the damage of an ore (Ali and Bradshaw, 2010). What the model calculations (Ali and Bradshaw, 2009, 2010; Jones et al., 2005) have in common is their focus on mineral comminution and that simulations are performed with idealistic two-phase rocks with one strongly absorbing mineral in a nonabsorbing matrix. Experimental studies of Satish et al. (2006), Peinsitt et al. (2010) show that also more homogeneous rocks such as basalt and granite show a significant increase in surface temperature as well as decrease in strength when irradiated in a multimode cavity. According to Peinsitt et al. (2010) basalt samples reach 330 °C after 60 s of irradiation with a power of 3.2 kW, whereas granite takes 300 s for reaching a temperature of 220 °C. The increase in temperature goes along with a significant decrease in uniaxial compressive strength and p-wave velocities reflecting a decrease in rock strength. The two authors demonstrate that relatively finely grained and homogeneous rocks can be heated and damaged with the help of microwaves. Therefore the presence of extremely good absorbing particles within this matrix working as a focus for microwave absorption and causing strong inhomogeneous heating is not a necessity. Nevertheless there is a strong need to getting a better understanding of the processes leading to microwave induced damage of rather homogeneous rocks. It is also necessary to back up
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simulations with experimental results. This work focuses strongly on fine grained basaltic rocks and their response to irradiation in a microwave cavity, the way damage occurs and the numerical simulation of laboratory results.
modulus G and Poisson’s ratio m are connected by the well known relations:
2. Experimental arrangement
All governing equations are formulated in a coordinate free manner. With the help of tensor calculus the equations can be written in any convenient coordinate system (Eringen, 1980). In this work cylindrical coordinates
Following the procedure described by Peinsitt et al. (2010) the microwave irradiation experiments were performed in a 3.2 kW multi-mode cavity of a commercial microwave oven at a frequency of 2.45 GHz. Cylindrical basalt samples, 50 mm of length and 50 mm of diameter are positioned in the center of the cavity atop of glassware being a very weak microwave absorber. For each load step a minimum number of five samples have been investigated for statistical reasons. The effect of microwave irradiation is evaluated (a) by measuring the temperature of irradiated samples by the help of an infrared thermometer and camera, (b) by measuring sound velocities (p-wave) along the z-axis of the samples, (c) by investigating resin-saturated thin-sections and (d) by the help of a penetration spray normally used in metallography displaying cracks developing at the surface of a sample. Before preparing a thin section the samples are saturated with a blue resin. This resin only penetrates open cracks, hardens and helps identifying cracks and structural damages within the sample. 3. Numerical analysis 3.1. Governing equations The governing equations are obtained from the fundamental principles of continuum thermomechanics. Starting from the energy-balance equation for a non-moving body the heat conduction equation
qcp
@Tðr; tÞ ¼ r ðkrTðr; tÞÞ þ sðr; tÞ @t
ð1Þ
is obtained. The symbols in Eq. (1) are the temperature T the density q, the specific heat cp, the thermal conductivity k, the position vector r and the Nabla operator r. The symbol s denotes the heat generation rate in the medium, generally specified as heat generation per unit of time and per unit of volume. When SI-units are used, s is given in W/m3. In this work the thermal properties of the medium are temperature dependent, therefore Eq. (1) becomes a nonlinear parabolic partial differential equation. For the linear heat conduction equation several analytical solutions exist, while in the nonlinear case only for few special problems analytical solutions can be obtained (Carslaw and Jaeger, 1986; Ozisik, 1993). For this reason a numerical solution of Eq. (1) is performed. To solve the thermoelastic problem additional equations are required. These are the equilibrium equation
r r ¼ 0;
ð2Þ
1 2
3k þ 2l ; kþl
x ¼ r cos u;
G ¼ l;
y ¼ r sin u;
ð3Þ
and
m¼
k : 2ðk þ lÞ
ð5Þ
z¼z
ð6Þ
0 6 u 6 2p and 0 6 z 6 H;
ð7Þ
with
0 6 r 6 R;
are used. The temperature dependence of the thermal conductivity k and the specific heat cp are taken into account. In Section 3.3 the values of k(T) and cp(T) at different temperatures are given. The density of the material is constant. As mentioned before the symbol s in Eq. (1) denotes an internal heat generation rate (in W/m3) or a body flux. There exist several approaches to model microwave absorption in dielectric media. For example Dincov et al. (2004) utilize Maxwell’s equations in dielectric media, or Ni et al. (1999) use Lambert–Beer’s law to model the absorption which allows to calculate the heating. In this work a simplified approach is performed. It is assumed that s is constant throughout the whole volume. This is justified by (1) the sample being irradiated in the microwave oven from all directions and (2) the penetration depth Dp being larger than the sample dimensions. For average values of j0 and j00 (7.4 and 0.48, resp., Santamarina, 1989) Dp is 11 cm, which has to be compared with the radius and half the height of the cylindrical sample (2.5 cm). Based on thermograhic measurements the heat generation rate s is calibrated with aid of the FE-method. 3.2. Model There exist two Finite-element (FE)-models in this work, the first one is used to calibrate the volumetric heat source s, the second one is used to study the thermomechanical loading of the basalt sample. An analytical solution of the problem is not available, therefore it is solved numerically by the finite element method (FEM). As solver the general purpose FEM package ABAQUS v6.10 ABAQUS (2010) is used. The finite element discretization of the heat conduction equation is given by
_ þ ½KfTg ¼ fQ g ½CfTg
ð8Þ
with the heat capacity matrix [C], the conductivity matrix [K], the temperature vector {T} and the load vector {Q} which takes the source-term into account. Formulating Eq. (8) at the nth time level delivers:
½CfT_ n g þ ½KfT n g ¼ fQ n g
with the Cauchy stress r, the strain–displacement relations
e ¼ ðru þ urÞ
E¼l
ð9Þ
Due to stability reasons, the time derivative fT_n g is approximated by the backward difference
with the infinitesimal strain tensor e and the displacement vector u, and the thermoelastic constitutive relation
T tþDt T t T_ tþDt ¼ Dt
r ¼ ktrðeÞ1 þ 2le ð3k þ 2lÞa1ðT T 0 Þ
in the time-integration scheme, for details see ABAQUS (2010). In this paper Eq. (9) is nonlinear, i.e., [K] and [C] are temperaturedependent, therefore a Newton–Raphson algorithm is used to solve the nonlinear equations. Details can be found in the manual of the FE-code ABAQUS (2010) or in Bathe (1995).
ð4Þ
with the Lamé constants k and l, the coefficient of thermal expansion a and the temperature T0 which characterizes a stress-free reference state. The Lamé constants, the Young’s modulus E, the shear
ð10Þ
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3.2.1. Thermal model With this model and a comparison with experimental results the strength of the volumetric heat source s is determined. In the experimental arrangement the cylinder is cut along the axis and put together by a special tape to guarantee the thermal contact. After irradiation the parts are drawn apart and a picture is taken with an infrared camera. A detailed illustration of the model is given in Fig. 1. The analyzed region is a circular cylinder with radius R = 25 mm and height H = 50 mm. The material is homogeneous and several properties are temperature-dependent (details see Section 3.3). The load case is divided into two steps. In the first one, the cylinder is heated up for Dt1 = 60 s by the body flux s. In the second step, Dt2 = 3600 s, the body flux is switched off and the cool down sequence of the half-cylinder during the infrared measurement is investigated for 3600 s. To obtain a solution of the considered thermal problem, the initial condition and the boundary conditions have to be specified. At the time t = 0 s the temperature is given by T(r, z, u, 0) = 25 °C. In Fig. 1 the boundary conditions for both steps are shown. In step 1 free convection with T1 = 25 °C and a heat transfer coefficient of a = [20] W/m2 K appears on the cylindrical surface. Additionally a radiation condition with T1 = 25 °C and an emissivity of e = 0.8 is enforced. In step 2 the symmetry condition on the mid plane is changed to free convection with T1 = 25 °C, a = [20] W/m2 K and radiation with T1 = 25 °C, e = 0.8. The results of the thermal modeling and the comparison with experimental results are already shown in this section. This prepares the basis for the thermophysical calculations in the following chapters. A temperature plot from the lower left to the upper right corner of the cross section of the sample can be seen in Fig. 2. Additionally the cut sample and the picture from the infrared camera are shown. The strength of the volumetric heat source s is calibrated so that the maximum temperature in the center of the sample calculated by the FE-model is equal to the measured value. By this procedure a numerical value of [16.8 106] W/m3 is obtained for s. The difference between the measured and calculated curves in Fig. 2 is considered as being mainly due to two reasons: (1) The calculation assumes a constant body flux throughout the sample. In the experiment the interior becomes hotter. Consequently j00 increases there and the absorbed energy is higher than near the
Fig. 2. Thermographic image of the interior of a sample after 60 s of irradiation and subsequent cooling for 5 s (bottom left); Position of the trend line AB on the sample (bottom right); Measured and calculated temperature along trend line (curves).
surface. Absorbed energy is equivalent to body flux which therefore actually decreases towards the surface. Consequently the calculated curve would be changed in the direction of the experimental one. (2) In the experiment, near the surface also some contribution of the cooler surrounding is measured. This causes the steep decrease of the temperature in the outermost millimetres. A better agreement between experiment and simulation is provided in Fig. 3. Here heating and cooling curves measured at the surface of a sample are plotted for a microwave irradiation of 60 s and subsequent cooling for 14 min. Together with the arguments of the previous paragraph this justifies using the thermal model as the input for the thermomechanical model. 3.2.2. Thermomechanical model As in the analytical description the advantage of axial symmetry is also used in the FEM formulation of the problem. A detailed
Fig. 1. Thermal FE model, geometry, elements, boundary conditions, initial conditions.
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450
Measurement Computation
400 350
Table 1 Thermal conductivity k(T), after Cermak and Rybach (1982). T in °C k in W/mK
0 2.09
50 2.16
100 2.20
200 2.12
300 1.98
400 1.89
500 1.83
T in ºC
300 250
Table 2 Specific heat cp(T), after Bouhifd et al. (2007).
200
T in °C cp in J/kg K T in °C cp in J/kg K
150 100
117 813 452 950
197 859 527 968
247 881 587 980
292 899 637 992
342 914 687 1008
407 936 797 1025
427 940 887 1040
50 0 0
100
200
300
400
500
600
700
800
900
t in s Fig. 3. Comparison of measured and computed temperature T at the point P(25 mm, 25 mm).
illustration of the model is given in Fig. 4. As in the thermal model the analyzed region is a circular cylinder with radius R = 25 mm and height H = 50 mm. The material is homogeneous and several properties are temperature-dependent (details see Section 3.3). To obtain a solution of the considered initial-boundary value problem, the initial condition and the boundary conditions have to be specified. At the time t = 0 s the temperature is given by T(r, z, 0) = 25 °C. In Fig. 4 it is shown that on all boundaries free convection with T1 = 25 °C and a heat transfer coefficient of a = 20 W/ m2 K appears. Additionally on all boundaries a radiation condition with T1 = 25 °C and an emissivity of e = 0.8 is enforced. As mechanical boundary conditions the point P(0, H) is fixed (ur(0, H) = 0, uz (0, H) = 0) and all nodes on the axis of the cylinder are fixed in radial direction (ur(0, z) = 0). Additionally at T0 the circular cylinder is in a stress free state. The load case is divided into two steps. In the first one, the cylinder is heated up for Dt1 = 60 s by the body flux s(r, z, t). In the second step, Dt2 = 3600 s, the body flux is switched off and the cool down sequence of the cylinder is investigated for 3600 s.
3.3. Thermophysical Properties of basalt To analyze the thermomechanical problem several thermophysical properties are necessary. These parameters are the thermal conductivity k the specific heat cP, the density q. For the subsequent stress calculation the Young’s modulus E, the Poisson’s ratio m and the coefficient of thermal expansion a are required. Most of the properties are temperature-dependent, the values of all material parameters are listed in Tables 1–4. 4. Results 4.1. Microwave irradiation Figs. 5 and 6 show the results of microwave irradiation of basaltic samples, i.e. temperature and sound velocity, respectively, as function of irradiation time. In order to see the variations clearly the illustration with a box-plot diagram is used. The box contains 50% of all the values. The dark horizontal line indicates the median. Values that lie outside that box are covered by the so-called whiskers whose length is maximally 1.5 times the height of the box. Outliers and extreme outliers are outside the range of the whiskers. They are defined via the range they cover relative to the height of the box, i.e., 1.5–3 times the height (outliers), more than three times (extreme outliers). It can be seen that the surface temperature increases with increasing irradiation time and thus energy
Fig. 4. Thermomechanical FE model, geometry, meshsize, elements, boundary conditions, initial conditions.
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Table 3 Young’s Modulus E(T) and Poisson’s ratio m, after Peinsitt (2009). T in °C E in GPa m in 1
82 24 0.22
148 27 0.22
214 32 0.22
280 29 0.22
413 20 0.22
Table 4 Density q, coefficient of thermal expansion a, after Cermak and Rybach (1982).
q in kg/m3 a in 1/K
2.8 103 8 106
Fig. 6. Sound velocity (vp) [m/s] of basalt samples as function of irradiation time [s].
Fig. 5. Surface temperature [°C] of basalt samples after irradiation with 3.2 kW as function of irradiation time [s].
input. Surface temperature reaches 110 °C after 30 s, 280 °C after 60 s and 450 °C after 120 s of irradiation. This goes along with a significant decrease in sound velocity along the sample axis from approximately 5500 m/s in the initial, untreated state down to 3500 m/s after 120 s of irradiation. It is well known that heating with microwaves does not primarily happen at the surface of a sample but mainly in the interior of a piece of rock. The penetration depth and heating rates are a function of wavelength and dielectric properties of the samples. The temperature distribution within a sample can be inhomogeneous and vary with time depending on (a) the penetration and absorption of microwaves and (b) the heat flux. Due to the fact that every piece of metal acts as antenna and disturbs the field distribution the insertion of thermocouples into microwave cavities is problematic. So a good way to getting a better insight into temperature distribution within a sample is by means of infrared cameras. A sample was cut along the cylinder axis and put together by adhesive tape again. In this way the thermal contact between the two parts could be guaranteed and the influence on the heat transfer at this plane is considered to be minimal. Additionally the parts can be easily drawn apart after irradiation and a picture taken with the infrared camera. In this way an insight into temperature distribution within the samples can be given. After 60 s of irradiation a basalt sample has a surface temperature of approximately 250 °C (Fig. 5). A picture taken with an infrared camera approximately 5 s after the end of the irradiation for 60 s shows that the temperature increases towards the center of the sample up to 440 °C (Fig. 2, down left). It can be seen that the temperature decreases uniformly towards the rims which are cooler by approximately
200 °C than the centers. A temperature trend from lower left side to the upper right corner of the sample can be seen in Fig. 2 (above). Development of stresses is going along with the heating of a sample. Whereas samples show no cracks before being irradiated with microwaves (Fig. 7), large cracks develop when irradiating them. Fig. 8 shows a basalt sample after 60 s of irradiation with 3.2 kW. A penetration spray is used to get better visibility of the cracks. Cracks develop mainly parallel to the cylinder axis of the sample as well as at its circumference. Another set of radial cracks is visible at the front end. Cutting the sample along the axis and preparing a resin saturated thin-section reveals a situation as shown in Fig. 9. Two major directions of cracks are displayed (blue)1; (a) parallel to the long axis and (b) in radial direction. Cracks develop perpendicular to each other and cross-cut the mineralogical components of the sample. The two major cracks displayed in Fig. 9 are crosscutting small particles as well as larger pyroxene grains. A link of these cracks to any grain boundary cannot be observed. 4.2. Thermomechanical calculations For both load steps, Dt1 = 60 s (heating period) and Dt2 = 3600 s (cooling period), the temperature field T(r, z, t) and the stress field r(r, z, t) are calculated. Some results are shown in the following diagrams and contour plots. In Fig. 10 the temperature of the cylinder at t1 = 60 s is shown. Due to the symmetry of the problem, the temperature field is symmetric about the line z = H/2, i.e., the maximum temperature occurs at z = H/2. A plot of T along the specimen’s axis is given by Fig. 11. The components of the stress tensor r at t = 60 s are shown in Fig. 12. In Fig. 12a the radial stress rr(r, z, 60 s), in Fig. 12b the axial stress rz(r, z, 60 s) and in Fig. 12c the circumferential stress ru(r, z, 60 s) is plotted. If for basalt rock a typical tensile strength of rmax = 9 MPa is assumed (Hustrulid et al., 2001), it can be seen that this limit is exceeded in several regions of the cylinder. Exact values for the time-dependent development of stress of point P(25 mm, 25 mm) as well as the maximum tensile strength can be seen in Fig. 13. Therefore, during the thermomechanical loading damage can occur 1 For interpretation of color in Figs. 1–4 and 7–13, the reader is referred to the web version of this article.
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Fig. 7. Basalt sample treated with penetration spray before irradiation with microwaves.
Fig. 10. Temperature T(r, z, 60s).
and microcracks can be initiated. Due to the subsequent loading these cracks become macroscopic cracks. In this figure it can be seen that at the end of the heating phase and at the beginning of the cooling phase the tensile stress is exceeding tensile strength indicating the onset of damage. 5. Discussion Fig. 8. Basalt sample treated with penetration spray after irradiation with 3.2 kW microwave for 60 s.
Several publications deal with microwave irradiation of rocks and the damage going along with it. What all these papers have
Fig. 9. Thin section taken from the center of basalt-sample after irradiation for 60 s with 3.2 kW. Cracks saturated with blue resin.
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Fig. 13. Components of the stress tensor r at the point P(25 mm, 25 mm).
Fig. 11. Temperature T(0, z) along the axis of the specimen.
in common is their devotion to the irradiation of relatively small particles within a fine grained matrix and a clear separation of large absorbing and fine non-absorbing phases (Whittles et al., 2003; Jones et al., 2005, 2007; Ali and Bradshaw, 2009). None of these works deals with the irradiation of large samples with a fine grained composition. Components of the rock type presented in the present work are mostly <100 lm in size. The observation that cracks cross-cut mineralogical boundaries (Fig. 9) is a key finding. No damage in direct association with the mineralogy like cracks along grain boundaries can be found. This means that a fine grained rock like basalt does not show effects of differential heating and thermal expansion of individual grains when the microwave absorption process is slow. Slow means that the heating rate is small compared to the heat
transfer rate between grains and between the sample and surrounding medium. This impression is supported by taking a look at a larger scale. Here one can clearly see that cracks are aligned parallel to the axis of the sample and along its circumference (Fig. 8). The fine grained structure of basalt and the smoothness of the temperature distribution (Fig. 2) are the justification that the sample could be described in the simulations by a homogeneous model. The strength of the volumetric heat source was calibrated by the help of a thermal model. In this work the occurring stresses are computed with thermophysical properties taken from various literature sources. It is well known that these values show a very broad variation. This causes some uncertainties in numerical simulations and may be the root for discrepancies between experiment and simulation. Heating up of samples is observed to be stronger in the center (440 °C compared to 300 °C near the surface; see Fig. 2). As long as microwaves are absorbed by the whole sample equally, heating
Fig. 12. Components of the stress tensor r at t = 60 s in Pa, (a) radial stress rr(r, z, 60 s), (b) axial stress rz(r, z, 60 s), (c) circumferential stress ru(r, z, 60 s).
P. Hartlieb et al. / Minerals Engineering 31 (2012) 82–89
should be equally distributed. Heat transfer and radiation leads to cooling of the rims. On the other hand thermal conduction is not fast enough to compensate for this loss. The result is a higher temperature in the center than at the rims leading to an increased j00 and thus better heating of the inner (hotter) parts. This effect can even result in a thermal runaway (Peinsitt et al., 2010). In that work temperatures as high as 1220 °C and melting of a sample’s interior was observed when irradiated for 260 s with 3.2 kW. The result is an increased temperature gradient with longer irradiation times. 6. Conclusion This paper is about experimental and numerical investigations of microwave heating in cylindrical basalt samples. With an applied power of 3.2 kW in a multi-mode cavity temperature reached up to 250 °C at the surface and 440 °C in the center of cylindrical samples after 60 s. The samples show a multitude of cracks aligned parallel to their axes and in a radial direction. Sound velocities are reduced from approximately 5500 m/s in the initial, untreated state down to 3500 m/s after 120 s of irradiation. The thermomechanical FE-model introduced in this work is found to be a useful tool to estimate the induced thermal stresses and the conditions for the onset of damage and formation of cracks in rock samples with dimensions of the order of the temperature gradients developing under microwave irradiation. A necessary input in this thermomechanical model is the volumetric heat source (or body flux) which is determined by a thermal FE-model and by comparison with experimental results of the temperature distribution in the sample. To study the evolution of damage and crack formation, more complex constitutive relations are required. Also the effect of the cracks on heat conduction should not be disregarded. Additionally, such a problem can be analyzed only with the help of computational fracture mechanics which will be the subject of future research along with the determination of thermophysical rock properties for the investigated rock types. Acknowledgements We thank N. Sifferlinger, H. Kargl, M. Gimpel and Uwe Restner for valuable discussions. Sandvik Mining and Construction provided equipment and sample material. This work was supported by the ‘‘FFG – Austrian Research Promotion Agency’’ (Project Nr.: 820125/18797).
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References ABAQUS, 2010. ABAQUS v6.10 Documentation. Dassault Systèmes Simulia Corp., Providence, RI, USA. Ali, Y.A., Bradshaw, M.S., 2009. Quantifying damage around grain boundaries in microwave treated ores. Chemical Engineering and Processing: Process Intensification 48 (11–12), 1566–1573. Ali, Y.A., Bradshaw, M.S., 2010. Bonded-particle modelling of microwave-induced damage in ore particles. Minerals Engineering 23 (10), 780–790. Bathe, K., 1995. Finite Element Procedures, 2nd ed. Prentice Hall, Inc., Englewood Cliffs, New Jersey. Bouhifd, M.A., Besson, P., Courtial, P., Gérardin, C., Navrotsky, A., Richet, P., 2007. Thermochemistry and melting properties of basalt. Contrib. Mineral. Petrol. 153 (6), 689–698. Carslaw, H., Jaeger, J., 1986. Conduction of Heat in Solids, second ed. Oxford University Press, Oxford. Cermak, V., Rybach, L., 1982. Table 7 (igneous – metamorphic rocks). In: Angenheister, G. (Ed.), Landolt-Börnstein – Group V Geophysics, vol. c. Springer-Verlag, Berlin/Heidelberg, pp. 315–325. Dincov, D.D., Parrott, K.A., Pericleous, K.A., 2004. Heat and mass transfer in twophase porous materials under intensive microwave heating. Journal of Food Engineering 65, 403–412. Eringen, A.C., 1980. Mechanics of Continua, second ed. Krieger Publishing Company, Huntington, New York. Hustrulid, W., McCarter, M., Van Zyl, D., 2001. Slope Stability in Surface Mining. Society for Mining, Metallurgy and Exploration. Jones, A.D., Kingman, W.S., Whittles, N.D., Lowndes, S.I., 2005. Understanding microwave assisted breakage. Minerals Engineering 18 (7), 659–669. Jones, A.D., Kingman, W.S., Whittles, N.D., Lowndes, S.I., 2007. The influence of microwave energy delivery method on strength reduction in ore samples. Chemical Engineering and Processing 46 (4), 291–299. Ni, H., Datta, A.K., Torrance, K.E., 1999. Moisture transport in intensive microwave heating of biomaterials: a multiphase porous media model. International Journal of Heat and Mass Transfer 42, 1501–1512. Ozisik, M.N., 1993. Heat Conduction, second ed. John Wiley & Sons, Inc., New York. Peinsitt, T., 2009. Demonstration of the feasibility of weakening rocks with microwave techniques. Master’s thesis, Graz University of Technology, Graz. Peinsitt, T., Kuchar, F., Hartlieb, P., Moser, P., Kargl, H., Restner, U., Sifferlinger, N., 2010. Microwave heating of dry and water saturated basalt, granite and sandstone. International Journal of Mining and Mineral Engineering 2 (1), 18– 29. Pickles, C.A., 2009. Microwaves in extractive metallurgy: Part 2 – A review of applications. Minerals Engineering 22 (13), 1112–1118. Santamarina, C.J. (Ed.), 1989. Rock Excavation with Microwaves: A Literature Review. Foundation Engineering: Current Principles and Practices. ASCE, Evanston, IL, USA. Satish, H., Quellet, J., Raghavan, V., Radziszewski, P., 2006. Investigating microwave assisted rock breakage for possible space mining applications. Transactions of the Institutions of Mining and Metallurgy, Section A: Mining Technology 115 (1), 34–40. Walkiewicz, W.J., Kazonich, G., McGill, L.S., 1988. Microwave heating characteristics of selected minerals and compounds. Minerals and Metallurgical Processing 5 (1), 39–42. Whittles, N.D., Kingman, W.S., Reddish, J.D., 2003. Application of numerical modelling for prediction of the influence of power density on microwaveassisted breakage. International Journal of Mineral Processing 68 (1–4), 71–91.
Contents lists available at SciVerse ScienceDirect
Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
Damage of basalt induced by microwave irradiation P. Hartlieb a,⇑, M. Leindl b, F. Kuchar c, T. Antretter b, P. Moser a a
Lehrstuhl für Bergbaukunde, Bergtechnik und Bergwirtschaft, Montanuniversität Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria Institut für Mechanik, Montanuniversität Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria c Institut für Physik, Montanuniversität Leoben, Franz-Josef-Straße 18, A-8700 Leoben, Austria b
a r t i c l e
i n f o
Article history: Available online 16 February 2012 Keywords: Microwave irradiation Finite elements Rock heating and damage
a b s t r a c t In this work microwave irradiation on cylindrical samples of basaltic rock is investigated by laboratory experiments and compared with results from numerical models. Due to the temperature gradient in the samples induced by the microwave irradiation a significant damage indicated by a reduction of the sound velocity and finally the formation of cracks occurs. Applying a microwave power of 3.2 kW leads to a surface temperature of 250 °C and to 400 °C in the centre of a cylindrical sample after 60 s of irradiation. Temperature rise goes along with the formation of both axial and radial cracks. Cracks are not bound to the mineralogical composition but their development is governed by macroscopic temperature gradients and the geometry of the sample. A thermal and a thermomechanical finite element model are formulated and used to calculate temperature distributions and induced thermal stresses. The results indicate that tensile stresses exceed tensile strength leading to cracks as observed experimentally. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Weakening rocks by means of microwave irradiation has been a major topic in research for the past 50 years. The focus of all the projects lies in the development of more economic technologies for breaking, cutting and comminution of various types of rocks. A detailed review of all the work done before the 1980s is provided by Santamarina (1989). Recent activities mainly focus on comminution of various rock types. They have been performed by Walkiewicz et al. (1988), Satish et al. (2006), Jones et al. (2005) and many more. Pickles (2009) gives a detailed overview possible applications in extractive metallurgy. The absorption of microwave energy of a sample depends on high-frequency dielectric properties of the constituents of a rock. These can be described by = 0 + i00 = 0(j0 + ij00 ). Where j0 is the real part, j00 the imaginary part of the relative dielectric constant and 0 is the permittivity of free space. According to Santamarina (1989) typical values for rocks range from 103–50 (j00 ) and 2–10 (j0 ). Values for j00 of basalt range from 0.08 to 0.8, whereas values for j0 vary between 5.4 and 9.4. Several papers are devoted to explaining rock breakage by the different microwave absorption rates and thus different heating rates of the mineralogical constituents of a sample. Whittles et al. (2003), Jones et al. (2005) postulate that inducing fractures between an ore and the host rock can be achieved by using the differential absorption of microwave energy between the different ⇑ Corresponding author. E-mail address: [email protected] (P. Hartlieb). 0892-6875/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mineng.2012.01.011
mineral phases. Together with Jones et al. (2007), Ali and Bradshaw (2009) it is demonstrated that damage after microwave-treatment is caused by tensile stresses occurring during the thermal expansion of the absorbing phases as well as shear stresses along the grain boundaries. It is also shown that particle size as well as delivery method and power density play a major role in the damage of an ore (Ali and Bradshaw, 2010). What the model calculations (Ali and Bradshaw, 2009, 2010; Jones et al., 2005) have in common is their focus on mineral comminution and that simulations are performed with idealistic two-phase rocks with one strongly absorbing mineral in a nonabsorbing matrix. Experimental studies of Satish et al. (2006), Peinsitt et al. (2010) show that also more homogeneous rocks such as basalt and granite show a significant increase in surface temperature as well as decrease in strength when irradiated in a multimode cavity. According to Peinsitt et al. (2010) basalt samples reach 330 °C after 60 s of irradiation with a power of 3.2 kW, whereas granite takes 300 s for reaching a temperature of 220 °C. The increase in temperature goes along with a significant decrease in uniaxial compressive strength and p-wave velocities reflecting a decrease in rock strength. The two authors demonstrate that relatively finely grained and homogeneous rocks can be heated and damaged with the help of microwaves. Therefore the presence of extremely good absorbing particles within this matrix working as a focus for microwave absorption and causing strong inhomogeneous heating is not a necessity. Nevertheless there is a strong need to getting a better understanding of the processes leading to microwave induced damage of rather homogeneous rocks. It is also necessary to back up
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simulations with experimental results. This work focuses strongly on fine grained basaltic rocks and their response to irradiation in a microwave cavity, the way damage occurs and the numerical simulation of laboratory results.
modulus G and Poisson’s ratio m are connected by the well known relations:
2. Experimental arrangement
All governing equations are formulated in a coordinate free manner. With the help of tensor calculus the equations can be written in any convenient coordinate system (Eringen, 1980). In this work cylindrical coordinates
Following the procedure described by Peinsitt et al. (2010) the microwave irradiation experiments were performed in a 3.2 kW multi-mode cavity of a commercial microwave oven at a frequency of 2.45 GHz. Cylindrical basalt samples, 50 mm of length and 50 mm of diameter are positioned in the center of the cavity atop of glassware being a very weak microwave absorber. For each load step a minimum number of five samples have been investigated for statistical reasons. The effect of microwave irradiation is evaluated (a) by measuring the temperature of irradiated samples by the help of an infrared thermometer and camera, (b) by measuring sound velocities (p-wave) along the z-axis of the samples, (c) by investigating resin-saturated thin-sections and (d) by the help of a penetration spray normally used in metallography displaying cracks developing at the surface of a sample. Before preparing a thin section the samples are saturated with a blue resin. This resin only penetrates open cracks, hardens and helps identifying cracks and structural damages within the sample. 3. Numerical analysis 3.1. Governing equations The governing equations are obtained from the fundamental principles of continuum thermomechanics. Starting from the energy-balance equation for a non-moving body the heat conduction equation
qcp
@Tðr; tÞ ¼ r ðkrTðr; tÞÞ þ sðr; tÞ @t
ð1Þ
is obtained. The symbols in Eq. (1) are the temperature T the density q, the specific heat cp, the thermal conductivity k, the position vector r and the Nabla operator r. The symbol s denotes the heat generation rate in the medium, generally specified as heat generation per unit of time and per unit of volume. When SI-units are used, s is given in W/m3. In this work the thermal properties of the medium are temperature dependent, therefore Eq. (1) becomes a nonlinear parabolic partial differential equation. For the linear heat conduction equation several analytical solutions exist, while in the nonlinear case only for few special problems analytical solutions can be obtained (Carslaw and Jaeger, 1986; Ozisik, 1993). For this reason a numerical solution of Eq. (1) is performed. To solve the thermoelastic problem additional equations are required. These are the equilibrium equation
r r ¼ 0;
ð2Þ
1 2
3k þ 2l ; kþl
x ¼ r cos u;
G ¼ l;
y ¼ r sin u;
ð3Þ
and
m¼
k : 2ðk þ lÞ
ð5Þ
z¼z
ð6Þ
0 6 u 6 2p and 0 6 z 6 H;
ð7Þ
with
0 6 r 6 R;
are used. The temperature dependence of the thermal conductivity k and the specific heat cp are taken into account. In Section 3.3 the values of k(T) and cp(T) at different temperatures are given. The density of the material is constant. As mentioned before the symbol s in Eq. (1) denotes an internal heat generation rate (in W/m3) or a body flux. There exist several approaches to model microwave absorption in dielectric media. For example Dincov et al. (2004) utilize Maxwell’s equations in dielectric media, or Ni et al. (1999) use Lambert–Beer’s law to model the absorption which allows to calculate the heating. In this work a simplified approach is performed. It is assumed that s is constant throughout the whole volume. This is justified by (1) the sample being irradiated in the microwave oven from all directions and (2) the penetration depth Dp being larger than the sample dimensions. For average values of j0 and j00 (7.4 and 0.48, resp., Santamarina, 1989) Dp is 11 cm, which has to be compared with the radius and half the height of the cylindrical sample (2.5 cm). Based on thermograhic measurements the heat generation rate s is calibrated with aid of the FE-method. 3.2. Model There exist two Finite-element (FE)-models in this work, the first one is used to calibrate the volumetric heat source s, the second one is used to study the thermomechanical loading of the basalt sample. An analytical solution of the problem is not available, therefore it is solved numerically by the finite element method (FEM). As solver the general purpose FEM package ABAQUS v6.10 ABAQUS (2010) is used. The finite element discretization of the heat conduction equation is given by
_ þ ½KfTg ¼ fQ g ½CfTg
ð8Þ
with the heat capacity matrix [C], the conductivity matrix [K], the temperature vector {T} and the load vector {Q} which takes the source-term into account. Formulating Eq. (8) at the nth time level delivers:
½CfT_ n g þ ½KfT n g ¼ fQ n g
with the Cauchy stress r, the strain–displacement relations
e ¼ ðru þ urÞ
E¼l
ð9Þ
Due to stability reasons, the time derivative fT_n g is approximated by the backward difference
with the infinitesimal strain tensor e and the displacement vector u, and the thermoelastic constitutive relation
T tþDt T t T_ tþDt ¼ Dt
r ¼ ktrðeÞ1 þ 2le ð3k þ 2lÞa1ðT T 0 Þ
in the time-integration scheme, for details see ABAQUS (2010). In this paper Eq. (9) is nonlinear, i.e., [K] and [C] are temperaturedependent, therefore a Newton–Raphson algorithm is used to solve the nonlinear equations. Details can be found in the manual of the FE-code ABAQUS (2010) or in Bathe (1995).
ð4Þ
with the Lamé constants k and l, the coefficient of thermal expansion a and the temperature T0 which characterizes a stress-free reference state. The Lamé constants, the Young’s modulus E, the shear
ð10Þ
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3.2.1. Thermal model With this model and a comparison with experimental results the strength of the volumetric heat source s is determined. In the experimental arrangement the cylinder is cut along the axis and put together by a special tape to guarantee the thermal contact. After irradiation the parts are drawn apart and a picture is taken with an infrared camera. A detailed illustration of the model is given in Fig. 1. The analyzed region is a circular cylinder with radius R = 25 mm and height H = 50 mm. The material is homogeneous and several properties are temperature-dependent (details see Section 3.3). The load case is divided into two steps. In the first one, the cylinder is heated up for Dt1 = 60 s by the body flux s. In the second step, Dt2 = 3600 s, the body flux is switched off and the cool down sequence of the half-cylinder during the infrared measurement is investigated for 3600 s. To obtain a solution of the considered thermal problem, the initial condition and the boundary conditions have to be specified. At the time t = 0 s the temperature is given by T(r, z, u, 0) = 25 °C. In Fig. 1 the boundary conditions for both steps are shown. In step 1 free convection with T1 = 25 °C and a heat transfer coefficient of a = [20] W/m2 K appears on the cylindrical surface. Additionally a radiation condition with T1 = 25 °C and an emissivity of e = 0.8 is enforced. In step 2 the symmetry condition on the mid plane is changed to free convection with T1 = 25 °C, a = [20] W/m2 K and radiation with T1 = 25 °C, e = 0.8. The results of the thermal modeling and the comparison with experimental results are already shown in this section. This prepares the basis for the thermophysical calculations in the following chapters. A temperature plot from the lower left to the upper right corner of the cross section of the sample can be seen in Fig. 2. Additionally the cut sample and the picture from the infrared camera are shown. The strength of the volumetric heat source s is calibrated so that the maximum temperature in the center of the sample calculated by the FE-model is equal to the measured value. By this procedure a numerical value of [16.8 106] W/m3 is obtained for s. The difference between the measured and calculated curves in Fig. 2 is considered as being mainly due to two reasons: (1) The calculation assumes a constant body flux throughout the sample. In the experiment the interior becomes hotter. Consequently j00 increases there and the absorbed energy is higher than near the
Fig. 2. Thermographic image of the interior of a sample after 60 s of irradiation and subsequent cooling for 5 s (bottom left); Position of the trend line AB on the sample (bottom right); Measured and calculated temperature along trend line (curves).
surface. Absorbed energy is equivalent to body flux which therefore actually decreases towards the surface. Consequently the calculated curve would be changed in the direction of the experimental one. (2) In the experiment, near the surface also some contribution of the cooler surrounding is measured. This causes the steep decrease of the temperature in the outermost millimetres. A better agreement between experiment and simulation is provided in Fig. 3. Here heating and cooling curves measured at the surface of a sample are plotted for a microwave irradiation of 60 s and subsequent cooling for 14 min. Together with the arguments of the previous paragraph this justifies using the thermal model as the input for the thermomechanical model. 3.2.2. Thermomechanical model As in the analytical description the advantage of axial symmetry is also used in the FEM formulation of the problem. A detailed
Fig. 1. Thermal FE model, geometry, elements, boundary conditions, initial conditions.
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450
Measurement Computation
400 350
Table 1 Thermal conductivity k(T), after Cermak and Rybach (1982). T in °C k in W/mK
0 2.09
50 2.16
100 2.20
200 2.12
300 1.98
400 1.89
500 1.83
T in ºC
300 250
Table 2 Specific heat cp(T), after Bouhifd et al. (2007).
200
T in °C cp in J/kg K T in °C cp in J/kg K
150 100
117 813 452 950
197 859 527 968
247 881 587 980
292 899 637 992
342 914 687 1008
407 936 797 1025
427 940 887 1040
50 0 0
100
200
300
400
500
600
700
800
900
t in s Fig. 3. Comparison of measured and computed temperature T at the point P(25 mm, 25 mm).
illustration of the model is given in Fig. 4. As in the thermal model the analyzed region is a circular cylinder with radius R = 25 mm and height H = 50 mm. The material is homogeneous and several properties are temperature-dependent (details see Section 3.3). To obtain a solution of the considered initial-boundary value problem, the initial condition and the boundary conditions have to be specified. At the time t = 0 s the temperature is given by T(r, z, 0) = 25 °C. In Fig. 4 it is shown that on all boundaries free convection with T1 = 25 °C and a heat transfer coefficient of a = 20 W/ m2 K appears. Additionally on all boundaries a radiation condition with T1 = 25 °C and an emissivity of e = 0.8 is enforced. As mechanical boundary conditions the point P(0, H) is fixed (ur(0, H) = 0, uz (0, H) = 0) and all nodes on the axis of the cylinder are fixed in radial direction (ur(0, z) = 0). Additionally at T0 the circular cylinder is in a stress free state. The load case is divided into two steps. In the first one, the cylinder is heated up for Dt1 = 60 s by the body flux s(r, z, t). In the second step, Dt2 = 3600 s, the body flux is switched off and the cool down sequence of the cylinder is investigated for 3600 s.
3.3. Thermophysical Properties of basalt To analyze the thermomechanical problem several thermophysical properties are necessary. These parameters are the thermal conductivity k the specific heat cP, the density q. For the subsequent stress calculation the Young’s modulus E, the Poisson’s ratio m and the coefficient of thermal expansion a are required. Most of the properties are temperature-dependent, the values of all material parameters are listed in Tables 1–4. 4. Results 4.1. Microwave irradiation Figs. 5 and 6 show the results of microwave irradiation of basaltic samples, i.e. temperature and sound velocity, respectively, as function of irradiation time. In order to see the variations clearly the illustration with a box-plot diagram is used. The box contains 50% of all the values. The dark horizontal line indicates the median. Values that lie outside that box are covered by the so-called whiskers whose length is maximally 1.5 times the height of the box. Outliers and extreme outliers are outside the range of the whiskers. They are defined via the range they cover relative to the height of the box, i.e., 1.5–3 times the height (outliers), more than three times (extreme outliers). It can be seen that the surface temperature increases with increasing irradiation time and thus energy
Fig. 4. Thermomechanical FE model, geometry, meshsize, elements, boundary conditions, initial conditions.
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Table 3 Young’s Modulus E(T) and Poisson’s ratio m, after Peinsitt (2009). T in °C E in GPa m in 1
82 24 0.22
148 27 0.22
214 32 0.22
280 29 0.22
413 20 0.22
Table 4 Density q, coefficient of thermal expansion a, after Cermak and Rybach (1982).
q in kg/m3 a in 1/K
2.8 103 8 106
Fig. 6. Sound velocity (vp) [m/s] of basalt samples as function of irradiation time [s].
Fig. 5. Surface temperature [°C] of basalt samples after irradiation with 3.2 kW as function of irradiation time [s].
input. Surface temperature reaches 110 °C after 30 s, 280 °C after 60 s and 450 °C after 120 s of irradiation. This goes along with a significant decrease in sound velocity along the sample axis from approximately 5500 m/s in the initial, untreated state down to 3500 m/s after 120 s of irradiation. It is well known that heating with microwaves does not primarily happen at the surface of a sample but mainly in the interior of a piece of rock. The penetration depth and heating rates are a function of wavelength and dielectric properties of the samples. The temperature distribution within a sample can be inhomogeneous and vary with time depending on (a) the penetration and absorption of microwaves and (b) the heat flux. Due to the fact that every piece of metal acts as antenna and disturbs the field distribution the insertion of thermocouples into microwave cavities is problematic. So a good way to getting a better insight into temperature distribution within a sample is by means of infrared cameras. A sample was cut along the cylinder axis and put together by adhesive tape again. In this way the thermal contact between the two parts could be guaranteed and the influence on the heat transfer at this plane is considered to be minimal. Additionally the parts can be easily drawn apart after irradiation and a picture taken with the infrared camera. In this way an insight into temperature distribution within the samples can be given. After 60 s of irradiation a basalt sample has a surface temperature of approximately 250 °C (Fig. 5). A picture taken with an infrared camera approximately 5 s after the end of the irradiation for 60 s shows that the temperature increases towards the center of the sample up to 440 °C (Fig. 2, down left). It can be seen that the temperature decreases uniformly towards the rims which are cooler by approximately
200 °C than the centers. A temperature trend from lower left side to the upper right corner of the sample can be seen in Fig. 2 (above). Development of stresses is going along with the heating of a sample. Whereas samples show no cracks before being irradiated with microwaves (Fig. 7), large cracks develop when irradiating them. Fig. 8 shows a basalt sample after 60 s of irradiation with 3.2 kW. A penetration spray is used to get better visibility of the cracks. Cracks develop mainly parallel to the cylinder axis of the sample as well as at its circumference. Another set of radial cracks is visible at the front end. Cutting the sample along the axis and preparing a resin saturated thin-section reveals a situation as shown in Fig. 9. Two major directions of cracks are displayed (blue)1; (a) parallel to the long axis and (b) in radial direction. Cracks develop perpendicular to each other and cross-cut the mineralogical components of the sample. The two major cracks displayed in Fig. 9 are crosscutting small particles as well as larger pyroxene grains. A link of these cracks to any grain boundary cannot be observed. 4.2. Thermomechanical calculations For both load steps, Dt1 = 60 s (heating period) and Dt2 = 3600 s (cooling period), the temperature field T(r, z, t) and the stress field r(r, z, t) are calculated. Some results are shown in the following diagrams and contour plots. In Fig. 10 the temperature of the cylinder at t1 = 60 s is shown. Due to the symmetry of the problem, the temperature field is symmetric about the line z = H/2, i.e., the maximum temperature occurs at z = H/2. A plot of T along the specimen’s axis is given by Fig. 11. The components of the stress tensor r at t = 60 s are shown in Fig. 12. In Fig. 12a the radial stress rr(r, z, 60 s), in Fig. 12b the axial stress rz(r, z, 60 s) and in Fig. 12c the circumferential stress ru(r, z, 60 s) is plotted. If for basalt rock a typical tensile strength of rmax = 9 MPa is assumed (Hustrulid et al., 2001), it can be seen that this limit is exceeded in several regions of the cylinder. Exact values for the time-dependent development of stress of point P(25 mm, 25 mm) as well as the maximum tensile strength can be seen in Fig. 13. Therefore, during the thermomechanical loading damage can occur 1 For interpretation of color in Figs. 1–4 and 7–13, the reader is referred to the web version of this article.
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Fig. 7. Basalt sample treated with penetration spray before irradiation with microwaves.
Fig. 10. Temperature T(r, z, 60s).
and microcracks can be initiated. Due to the subsequent loading these cracks become macroscopic cracks. In this figure it can be seen that at the end of the heating phase and at the beginning of the cooling phase the tensile stress is exceeding tensile strength indicating the onset of damage. 5. Discussion Fig. 8. Basalt sample treated with penetration spray after irradiation with 3.2 kW microwave for 60 s.
Several publications deal with microwave irradiation of rocks and the damage going along with it. What all these papers have
Fig. 9. Thin section taken from the center of basalt-sample after irradiation for 60 s with 3.2 kW. Cracks saturated with blue resin.
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Fig. 13. Components of the stress tensor r at the point P(25 mm, 25 mm).
Fig. 11. Temperature T(0, z) along the axis of the specimen.
in common is their devotion to the irradiation of relatively small particles within a fine grained matrix and a clear separation of large absorbing and fine non-absorbing phases (Whittles et al., 2003; Jones et al., 2005, 2007; Ali and Bradshaw, 2009). None of these works deals with the irradiation of large samples with a fine grained composition. Components of the rock type presented in the present work are mostly <100 lm in size. The observation that cracks cross-cut mineralogical boundaries (Fig. 9) is a key finding. No damage in direct association with the mineralogy like cracks along grain boundaries can be found. This means that a fine grained rock like basalt does not show effects of differential heating and thermal expansion of individual grains when the microwave absorption process is slow. Slow means that the heating rate is small compared to the heat
transfer rate between grains and between the sample and surrounding medium. This impression is supported by taking a look at a larger scale. Here one can clearly see that cracks are aligned parallel to the axis of the sample and along its circumference (Fig. 8). The fine grained structure of basalt and the smoothness of the temperature distribution (Fig. 2) are the justification that the sample could be described in the simulations by a homogeneous model. The strength of the volumetric heat source was calibrated by the help of a thermal model. In this work the occurring stresses are computed with thermophysical properties taken from various literature sources. It is well known that these values show a very broad variation. This causes some uncertainties in numerical simulations and may be the root for discrepancies between experiment and simulation. Heating up of samples is observed to be stronger in the center (440 °C compared to 300 °C near the surface; see Fig. 2). As long as microwaves are absorbed by the whole sample equally, heating
Fig. 12. Components of the stress tensor r at t = 60 s in Pa, (a) radial stress rr(r, z, 60 s), (b) axial stress rz(r, z, 60 s), (c) circumferential stress ru(r, z, 60 s).
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should be equally distributed. Heat transfer and radiation leads to cooling of the rims. On the other hand thermal conduction is not fast enough to compensate for this loss. The result is a higher temperature in the center than at the rims leading to an increased j00 and thus better heating of the inner (hotter) parts. This effect can even result in a thermal runaway (Peinsitt et al., 2010). In that work temperatures as high as 1220 °C and melting of a sample’s interior was observed when irradiated for 260 s with 3.2 kW. The result is an increased temperature gradient with longer irradiation times. 6. Conclusion This paper is about experimental and numerical investigations of microwave heating in cylindrical basalt samples. With an applied power of 3.2 kW in a multi-mode cavity temperature reached up to 250 °C at the surface and 440 °C in the center of cylindrical samples after 60 s. The samples show a multitude of cracks aligned parallel to their axes and in a radial direction. Sound velocities are reduced from approximately 5500 m/s in the initial, untreated state down to 3500 m/s after 120 s of irradiation. The thermomechanical FE-model introduced in this work is found to be a useful tool to estimate the induced thermal stresses and the conditions for the onset of damage and formation of cracks in rock samples with dimensions of the order of the temperature gradients developing under microwave irradiation. A necessary input in this thermomechanical model is the volumetric heat source (or body flux) which is determined by a thermal FE-model and by comparison with experimental results of the temperature distribution in the sample. To study the evolution of damage and crack formation, more complex constitutive relations are required. Also the effect of the cracks on heat conduction should not be disregarded. Additionally, such a problem can be analyzed only with the help of computational fracture mechanics which will be the subject of future research along with the determination of thermophysical rock properties for the investigated rock types. Acknowledgements We thank N. Sifferlinger, H. Kargl, M. Gimpel and Uwe Restner for valuable discussions. Sandvik Mining and Construction provided equipment and sample material. This work was supported by the ‘‘FFG – Austrian Research Promotion Agency’’ (Project Nr.: 820125/18797).
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