Calculation of Thermal Conductivity of Diorite Rocks and Their Modeling

Transcription

1 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 12 No: Calculation of Thermal Conductivity of Diorite Rocks and Their Modeling Sarfraz Hussain 1, Sikandar Saneen 2, Namra Ahmad 2, M.Arshad Javeed 3, Shahbaz Ali 1 Department of Physics, Quaid-e-Azam University, Islamabad Pakistan 2 Department of Chemistry, University of Sargodha, Sargodha Pakistan 3 Government Degree College Ali Pure Chattha, Gujranwala, Pakistan ABSTRACT The main object of this work is to estimate the thermal conductivity of consolidated specimens of diorite rocks. The sample of these rocks have been obtained from Shaewa Shahbaz Ghari Volconic Complex near Mardan in Pakistan. The thermo-physical properties like thermal conductivity and thermal diffusivity of the specimens have been simultaneously measured by transient plane source technique at normal temperature and pressure. The porosity and density are helpful in the modeling of thermal conductivity. Therefore American Society for Testing and Materials (ASTM) standards are applied for the measurement of porosity and other density-related parameters. The chemical composition of the samples is made with the help of X-ray fluorescence technique. The effective thermal conductivity of thirteen samples of diorite have been obtained by using different pre-existing empirical models with air as saturent. A model is also proposed for the estimation of thermal conductivity at normal temperature and pressure. The values of effective thermal conductivity from all models are in agreement with the experimental values within twenty percent. KEY WORDS: Diorite, porosity, density, thermal conductivity, thermal diffusivity Corresponding author: sh_attari79@yahoo.com 1. INTRODUCTION The knowledge of the thermal transport properties has become important with the widespread interest in thermal processes like estimation of surface temperature gradients in geo-thermal reservoirs, oil recovery process and in underground nuclear waste disposal sites. It is difficult and prolonged to make an exact measurement of thermal conductivity of rocks. In the presence of all environmental conditions like fluid saturation, temperature and pressure, the laboratory measurements are almost prohibitive in terms of time and expense. Therefore many attempts have been made to formulate some empirical models for the estimation of thermal conductivity of porous rocks. The igneous rocks can be classified on the basis of texture and mineralogical composition. On the basis of grain size, the igneous rocks are divided [21] into intrusive, volcanic, or fine grained and extrusive, plutonic, or coarse grained but on the basis of mineralogical composition they fall into felsic, intermediate, mafic and ultra mafic. Diorite lies in the volcanic intermediate category of the igneous rocks. In these rocks (diorite) the silica content ranges from fifty to sixty percent by volume. The present work deals with the thermal parameters of the thirteen samples of the diorite. These samples

2 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 12 No: have been taken from the Shehwa Shahbaz Ghari Volconic Complex situated 60 Km away from the Indus River in Pakistan. American Society for Testing and Materials (ASTM) standards has used to measure porosity and density parameters. The porosity of these samples varies from to 0.510% by volume. The chemical composition was done with the help of X-ray fluorescence technique whereas the thermal properties are measured with the help of transient plane source technique at normal temperature and pressure. If the temperature and pressure is kept constant, the thermal conductivity of the understudied rocks depends on factors like porosity, pore fluids, chemical composition and geometry. The aim of this work is to predict thermal conductivity of igneous rocks (diorite) with the help of more easily measured physical parameters like porosity, density, thermal conductivity of pore saturants and thermal conductivity of solid phase of rocks. 2. THERMAL CONDUCTIVITY PREDICTING MODELS When there is no direct measurement performed then thermal conductivity of rocks can be inferred by various well defined models. These models can be classified [23] into three categories. The first type of thermal conductivity models are known as mixing laws. These models have short applications because they do not account the structural properties of rocks. The second type is called empirical models in which more easily measurable physical parameters related to thermal conductivity are used and regression analyses are applied to the obtained data. These models also have limitations as they are applicable only to the particular type of rocks under study. Third type is the theoretical model which involves the mechanism of transfer of heat to the simplified geometries of rock-fluid system. Again here, the difficulty is the degree of simplification to get a required solution. With the most other thermo-physical properties, in-situ thermal conductivity may deviate significantly from the laboratory values. Even the effect of temperature and pressure and pore saturants are accounted. This problem is due to difficulty of defining the representative elementary volume for which sensible average for transport parameters can be defined. Therefore, a general expression to forecast the effective thermal conductivity is still underproduction. If we suppose that the constituent minerals with thermal conductivities λ i and the volume concentration V i are arranged in parallel in the non-porous rocks then the thermal conductivity λ s of the pure solid phase is: λ s = λ i V i V i (1) In the following section, the mixinglaws and empirical models are discussed only. 2.1 MIXING-LAW MODELS There are three most well known mixing-law models [23] explained as below WEIGHTED ARITHMETIC MEAN This model involves the parallel distribution of solid phase and fluid phase with respect to the direction of heat flow. It provides the maximum value of effective thermal conductivity given as:

3 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 12 No: λ e = Фλ f + (1- Ф) λs (2) Where λ e is the effective thermal conductivity, λ f is the thermal conductivity of fluid contained in the pore space and Ф is the fractional porosity WEIGHTED HARMONIC MEAN This law involves the perpendicular arrangement of components (pure solid phase and pore fluid) with respect to the direction of heat flow and gives the minimum value of effective thermal conductivity as: λ e = ( φ λ f + 1 φ λ s ) WEIGHTEDGEOMETRIC MEAN (3) This law provides better results as compared to the two above mentioned but has no physical backgrounds. It can be expressed as: λ e = λ s ( λ f λ s) Ф (4) Where all the symbols have the same meaning as defined above. Horai (1991) tested the results of the predictions from several mixing-laws on a remarkable set of data and found that most of mixing-law models were valid only for certain porosity range. 2.2 EMPIRICAL MODELS The empirical models can be developed by related more easily measurable physical parameters to the thermal conductivity along with some empirical exponents, constants or adjustable parameters. The values of these components can be determined by applying least-square fit method to the laboratory data. Some of the empirical models are discussed below SUGAWARA-YOSHIZAWA MODEL The Sugawara and Yoshizawa [6] model can be expressed as: λ e = (1- A) λ s + A λ f (5) Where A =({ 2n 2 n 1}{ 1 1 (1 +φ) n}) is an adjustable parameter and n (>0) is the empirical exponent which depends upon the porosity, shape, orientation and emissivity inside the pores Asad's Model Asad's model [2] is very similar to the weighted geometric mean model which is given as: λ e = λ s ( λ cφ f λ s) (6) Where c is the empirical exponent AURANGZEB'S MODEL Auranzeb proposed an empirical model [25] for the estimation of thermal conductivity of consolidated porous media in terms of easily measurable parameters as: 1 λ e = 1 λ s + mφ λ f (7)

4 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 12 No: Where m is the empirical coefficient whose value can be determined by using experimental values of thermal conductivity and corresponding values of porosity Ф and thermal conductivity of solid phase of rock by using the relation given as: m = λf ( ( 1 1 λ exp λ s ) (8) φ PANDE- CHOUDARY MODEL The model proposed by the Pande and Choudary [14] is expressed as below: λ e = F(0.6132)( λ s λ f ) 1/2 [ ξ f ⅔ ] for ξ f >0 (9) λ e = F(0.6132)( λ s λ f ) 1/2 [ ξ s ⅔ ] for ξ s >0 (10) Where ξ f = Ф 0.5, ξ s = 0.5- Ф and F is an empirical coefficient EXPONENTIAL DECAY EQUTION An exponential decay equation [12] can also be used to estimate the effective thermal conductivity of consolidated porous media at room temperature and pressure. This expression is given as below: λ e = λs e -z Ф( λs/ λf) (11) Where z is the empirical exponent whose value can be determined by using experimental values of the thermal conductivity and corresponding values of Ф and λ s as: z Ф( λ s / λ f ) = ln ( λs/λ exp ) (12) The proposed model which can also be applied for the prediction of thermal conductivity of rocks is given as follows. λ e = λ s (1 Ф)e - Ф H (13) Where H is the adjustable parameter which can be determined with the help of experimental thermal conductivity, porosity and thermal conductivity of solid phase as given below: H = Ф ln( λ exp /λs ) + ln Ф(λ f /λ s ) (14) The empirical coefficients, exponents or adjustable parameters may differ according to the suite of rocks. Therefore, the extrapolation of empirical models to suites of rocks other than those used in developing these models may not be adequate. 3. SOURCE OF DATA AND TECHNIQUE The samples of diorite rocks have been taken from the Volconic Complex in Mardan. These rocks were cut into rectangular shapes having approximate dimensions of 5.0 x 3.5 x 2.5 cm 3 for which the density-related parameters and porosity were measured. The chemical composition of the specimens was done by X-ray fluorescence technique. For density-related parameters, the specimens were dried at 110 º C in the furnace for two days. After cooling at room temperature for half an hour, the specimens were kept into desiccators. For mass measurements, a digital balance is used. Thermal conductivity can be measured by lot of steady state and non-steady techniques like divided bar and needle

5 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 12 No: probe methods. In this work, the transient plane source (TPS) technique [22] was used to measure the thermal conductivity of the specimens. It allows the measurements without any disturbance from the interfaces between the sensor and the bulk specimens. Also, it can measure [26] thermal conductivity, thermal diffusivity and heat capacity per unit volume. In this technique, a TPS-element is used both for constant heat source and a sensor of temperature. For data collection, the TPS-element is sandwiched between two specimen halves. When a sufficiently [27,28] large amount of direct current is passed through the TPS-element, its temperature changes consequently and there is a voltage drop across the TPS-element. By recording this voltage drop for a particular time interval, detailed information about thermal conductivity ( λ) and thermal diffusivity(κ) is obtained. The heat capacity per unit volume (ρc p ) can be calculated from the relation given below. ρc p = λ/ κ (15) Where ρ is the density of the samples. 4. RESULTS AND DISCUSSION The thermal properties of rocks depend [30] upon their structure, mineral composition, porosity, density, thermal conductivity of solid phase, temperature and pressure. Grain density (ρ s ), bulk density (ρ) and porosity are grouped as density related properties. The density of solid phase, true density, or grained density (ρ s ) were calculated [31] by using the relation: ρ s = ρ i V i V i (16) Where ρ i and V i are the true densities and volume fractions of constituent minerals. The density related properties are given in table IV. It is evident from this table the porosity ranges from to by volume %. The chemical composition of thirteen specimens is given in table I and it is clear that the silica content varies from to by mass%age. The thermal conductivity, thermal diffusivity and heat capacity per unit volume of the samples are depicted in table III. The thermal conductivity for air as saturant varies from to Wm -1 K -1. Thermal diffusivity ranges from to mm 2 and heat capacity per unit volume to M J.m -1 k -1. The thermal conductivity of solid phase λ s of each sample was calculated and found to be equal to 1.5 Wm -1 K -1 while thermal conductivity for pore fluid is taken as Wm -1 K -1 for air [33]. For diorite samples, using air as saturant in Sagawra-Yoshizawa model, the empirical exponent n is taken as 1 and for Asad's model the empirical exponent c was calculated and its mean value is taken as The value of m in Auranzeb's model was calculated as and for exponential decay expression, the value of exponential coefficient z is taken For our proposed model, the value of H was calculated as All the values of effective thermal conductivities which were predicted from the above mentioned models have been enlisted in table V. From this table, it is inferred that the Asad s model has an error of 15%. While Aurangzeb's model and the exponential decay model gives an error more than 10% and Sagawar-Yoshizawa have error of 11%. Our proposed model has an error within 10%. A graph is shown in fig 1 between fractional porosity and effective thermal conductivities predicted from different models. Also

6 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 12 No: fig 2 gives the picture of variation in the mass percentage of silica content with the thermal conductivity. 5. CONCULOSION The diorite samples have been characterized by mineral composition, porosity and density. The chemical composition was done by using X-ray fluorescence technique. The density parameters have been measured by applying ASTM standards. In this work, an effort is made to propose an empirical model with the help of experimental data for diorite samples. The thermal conductivity of diorites air-saturated samples has also been predicted by some pre-existing models. The predicted values of thermal conductivity obtained by proposed model is in agreement with the experimental data within 10%. It is also noted that the variation of silica (by mass%) has a prominent relation with the thermal conductivity which increases with increase in the in mass percentage of silica content. It is also noted that most of empirical relations and adjustable parameters are different from material to material and from saturant to saturant. ACKNOWLEDGEMENTS The author wishes to acknowledge Mr. Qamar, Mr. Shafiq Cheema, Mr. M. Raffi and Mr. Nazer Hussain for their kind favour and support. REFFERENCES 1. K. Lichtnecker, Z.Phys.27: 115 (1926) 2. Asaad, Ph.D. Dissertation (Univ. of California, Berkeley, 1995) 3. A. A. Babanov, Sov. Phys. Tech. Phys. 2:617 (1957). 4. W. D. Kingery, J. Am. Ceram. Soc. 42:617 (1959). 5. A. Sugawara and Yoshizawa, Australian J. Phys.14: 468 (1961). 6. A. Sugawara and Yoshizawa, J. Appl. Phys.33: 3135 (1964). 7. A. D. Brailsford and K. G. Major, Br. J. Appl. Phys. 15:313 (1964). 8. J. Huetz, Progress in Heat and Mass Transfer (Oxford,Pergamon, 1970). 9. E. Gomma,Ph.D. Dissertation (Univ. of Calfornia, Berkely,19973). 10. A.E. Beck, geophysics 41:133 (1976). 11. H. Ozbek, Ph. D. Dissertation (Univ. of Calfornia,Berkely, 1976). 12. Aurangzeb and A. Maqsood, Int. J. Thermophys. 28:1371 (2007). 13. A. Ghafari, Ph. D. Dissertaion (Univ. of California, Berkely,(1980). 14. R. N. Pande and D. R. Chaudary, Pramana 22:63 (1984). 15. R. W. Zimmerman, J. Pet. Sc. Eng. 3:219 (1989). 16. K. Misra, A. K. Shrotriya, R. Singh and D.R. Chaudary, J. Phys. D:Appl.Phys.27:732 (1994). 17. A. Bouguerra, J. P. Laurant, M.S. Goual and M. Queneudec, J. Phys.D: Appl. Phys. 30:2900 (1997). 18. K. J. Singh, R. Singh and D. R. Chaudary, J. Phys. D: Appl. Phys. 31:1631 (1998). 19. A. Bouguerra, J. Phys. D: Appl. Phys. 32:1407 (1999). 20. I. H. Gul and A. Maqsood, Int. J. Thermophys. 27: 614 (2006). 21. S. C. Hurlbut, Dana s Manual of Mineralogy (Jhon Wiley & Sons, New York, 1971). 22. S. E. Gustafsson, Rev.Sci. Instrum. 62:797 (1991). 23. W. H. Somerton, Thermal Properties and temperature related Behaviour of Rock/Fluid System. (Elsevier, New York, 1992). 24. W. Woodside and J. H. Messmer, J. Appl. Phys. 32:1688 (1961).

7 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 12 No: Auranzeb, Z. Ali, S. F. Gurmani and A. Maqsood, J. Phy.D: Appl. Phys. 39: 3876 (2006). 26. S. E. Gustafsson, E. Karawacki and M. N. Khan, J. Phys. D: Appl. Phys. 12:1411(1979). 27. A. Maqsood,N. Amin, M. Maqsood, G. Shabbir, A. Mahmood and S.E. Gustafsson Int. J. Energy Res. 18:777 (1994). 28. M. A. Rehman and A. Maqsood, J.Phys D: Appl. Phys. 35:2040 (2002). 29. M. Maqsood, M. Arshad, M. Zafarullah and A. Maqsood, Supercond. Sci. Technol. 9:321 (1996). 30. K.Horai and G. Simmons, Earth Planet Sci. Lett.6:359 (1969). 31. Y. S. Touloukian, W. R. Judd and R.F. Roy,Physical Properties of Rocks and Minerals (McGraw-Hill, New York, 1981). 32. K. Horai, J. Geophys.Res. 76:617 (1971). 33. Horai, J. Geophys.Res.96 (B3): 4125 (1991)

8 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 12 No: Table I. Chemical Composition of Diorite Samples (Mass %) Sample SiO 2 TiO 2 AlO 3 Fe 2 O 3 FeO MnO MgO CaO Na 2 O K 2 O P 2 O 5 SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D

9 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 12 No: Table II. Physical properties of Diorite Samples Specimen Dry Weight Suspended Saturated Volume Weight Weight D (gm) S (gm) W (gm) V(cm 3 ) SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D

10 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 12 No: Table III. Thermal properties of Diorite Samples at room temperature. Specime n Thermal Conductivity Thermal Diffusivity Κ (mm 2.sec -1 ) Specific Heat ρc p (MJ.m -1 K -1 ) λ (Wm -1 K -1 ) Mean St.Dev. Mean St.Dev. Mean St.Dev. SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D

11 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 12 No: Table IV. Density related properties of Diorite Samples Specimen Water Absorption Bulk Density(ρ) Apparent Specific Gravity(γ) Fraction Porosity (Ф) (%) g cm -3 SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D

12 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 12 No: Table V. Experimental (λ exp ) & effective (λ e ) Thermal Conductivities (W m -1 K -1 ) calculated according to different models at normal temperature and pressure using air as saturant. Specimen λ exp Sugawara- Yoshizawa Asaad s model Aurangzeb s Model Exponential Decay λ e = λ s (1-Ф)e - Ф H (Proposed) λ e % Dev λ e % Dev λ e % Dev λ e % Dev λ e % Dev SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D SSG-D

13 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 12 No: Thermal Conductivity (Wm -1 K -1 ) Experimental Thermal Conductivity Estimated Thermal Conductivity From Proposed Model Fractional Porosity Fig Thermal Conductivity (W m -1 K -1 ) Si 2 O (mass %) Fig.2