Thermophysical properties of dunite rocks as a function of temperature along with the prediction of effective thermal conductivity

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1 Vol.2, No.6, (200) Natual Science Themophysical popeties of dunite ocks as a function of tempeatue along with the pediction of effective themal conductivity Auang Zeb *, Tayyaba Fidous 2, Asghai Maqsood 3 Depatment of Applied Physics, Fedeal Udu Univesity of Ats Science and Technology, Islamabad, Pakistan; *Coesponding Autho: auangzebbaba@yahoo.com 2 Depatment of Physics, Ai Univesity, Islamabad, Pakistan; 3 Themal Tanspot Laboatoy, School of Chemical and Mateials Engineeing, NUST, Islamabad, Pakistan; asghaimaqsood@yahoo.com Received 29 Decembe 2009; evised 4 Mach 200; accepted 7 Apil 200. ABSTRACT The themal conductivity, themal diffusivity and heat capacity pe unit volume of dunite ocks taken fom Chillas nea Gilgit, Pakistan have been measued simultaneously using tansient plane souce technique. The tempeatue dependence of themal tanspot popeties is studied in the tempeatue ange K. Diffeent elations fo the estimation of themal conductivity ae also tested. Themal conductivity data obey the modified Eucken s law in the tempeatue ange of measuements. Keywods: Dunite; Density; Poosity; Themal Conductivity; Tansient Plane Souce (TPS) Technique. INTRODUCTION The most elevant themal paametes of ocks ae themal conductivity, heat capacity pe unit volume and themal diffusivity. The fist two paametes give the capability of a mateial to conduct and accumulate heat, espectively; and the last one epesents how fast heat diffuses though a mateial []... Intoduction to Samples Igneous ocks ae classified on the basis of textue and chemical composition. On the basis of textue igneous ocks ae classified in to two goups.... Extusive o Volcanic When magma comes out of the suface of eath (called lava) then apid cooling and cystallization of that ma- gma above the eath suface fom volcanic igneous ocks. Most types of lavas cool apidly, esulting in the fomation of ocks composed mainly of micoscopic cystals. Some lavas cool so quickly that they fom a smooth volcanic glass called obsidian. These ae too finegained o glassy that thei mineal composition cannot be obseved without the use of petogaphic micoscope...2. Intusive o Plutonic The cooling and cystallization of molten magma below the suface of eath fom these ocks. Magma that foms the intusive ocks solidifies elatively slow; and so, most of the intusive ocks have lage cystals than of extusive ocks. The mineal gain size of these ocks is visible to naked eye even. On the basis of chemisty igneous ocks ae classified in to fou goups [2]...3. Felsic Igneous ocks deived fom felsic magma contain elatively high quantities of sodium, aluminium and potassium and ae composed of moe than 65% silica (SiO 2 ). Some common felsic igneous ocks include fine-gained Rhyolite and coase-gained Ganite. All of felsic ocks ae light in colou because of the dominance of quatz, potassium and sodium feldspas, and plagioclase feldspas mineals...4. Intemediate Some igneous ocks having chemisty between felsic and mafic ocks ae known as intemediate. Silica amounts fom 52% to 65%. Andesite (intusive) and Dioite (extusive) ae intemediate igneous ocks. These ocks ae composed pedominantly Plagioclase feldspa, Amphibole and Pyoxene mineals...5. Mafic Igneous ocks deived fom mafic magma ich in cal-

2 A. Zeb et al. / Natual Science 2 (200) cium, ion and magnesium and elatively poo in silica, amounts fom 45% to 52%. Some common mafic igneous ocks include-fine gained Basalt and coase-gained Gabbo. Mafic igneous ocks tend to be dak in colou because they contain a lage popotion of mineals ich in ion and magnesium (Pyoxene, Amphiboles and Olivine)...6. Ultamafic These ocks contain elatively low amount of silica < 45% and ae dominated by the mineals olivine, calcium ich plagioclase feldspas and pyoxene. Peidotite and Dunite ae the most common ultamafic ocks having coase-gained textue. Thee is no known moden finegained ultamafic ock. The samples studied hee belong to the dunite goup of igneous ocks. The density elated popeties of the selected samples at oom tempeatue along with thei themal popeties in tempeatue ange ( K) ae aleady published [3]. In this pape only the themal tanspot popeties ae measued in tempeatue ange 83 to 303 K at nomal pessue, with ai as satuant in poe spaces, using Tansient Plane Souce Technique [4]. In continuation of ou pevious wok [5,6], the aim in this wok is to study themal tanspot popeties of dunite ocks as a function of tempeatue and to test vaious elations fo the pediction of effective themal conductivity of poous media. 2. EXPERIMENTAL TECHNIQUES AND SAMPLE CHARACTERIZATION The themal tanspot popeties of the samples wee measued as a function of tempeatue, using tansient plane souce (TPS) technique. The beauty of TPS technique is that it allows measuements without any distubance fom the intefaces between the senso and the bulk samples. Also, simultaneous measuement of themal conductivity and themal diffusivity is possible [4]. In this technique, a TPS-element (Figue ) sandwiched between two halves of the sample is used both as a constant heat souce and a senso of tempeatue. Sample Pieces Insulating Layes Connecting Leads TPS-Element Nikel Foil Figue. A TPS-senso sandwiched between sample pieces [7]. Fo data collection the TPS-element (20 mm diamete) was used in a bidge cicuit, shown in Figue 2. When sufficiently lage (constant) amount of diect cuent is passed though the TPS-element, its tempeatue changes consequently and thee is a voltage dop acoss the TPS-element. By ecoding this voltage dop fo a paticula time inteval, detailed infomation about the themal conductivity ( ) and themal diffusivity ( ) of the test specimen is obtained. The heat capacity pe unit volume ( C P ) can then be calculated fom the elation: C P, () The esults of the themophysical measuements on the samples at diffeent tempeatues and nomal pessue ae shown in Figue 3. It is to be noted that the esults of themal popeties in high tempeatue ange 303 to 483 K [3] ae also included in Figue 3, so that one can obseve the oveall behavio of themal popeties of these samples ove a wide ange of tempeatue ( K). Taking into consideation the eos of the technique [7,8], standad deviations of the measuements and the sampling eos, the themal conductivity and themal diffusivity data contain eos of 5% and 7% espectively. The eos in volumetic heat capacity ae aound 0%. 3. THERMAL CONDUCTIVITY PREDICTION A typical equation used fo tempeatue dependence of lattice (phonon) themal conductivity is: A BT, (2) ( T ) a R p c R R 2 R s TPS element d Powe Supply R 0 + b DVM HP-IB Figue 2. Bidge cicuit diagam fo TPS technique: R s = Standad esistance, R 0 = Resistance of TPS senso [9].

3 628 A. Zeb et al. / Natual Science 2 (200) Figue 3. Tempeatue dependence of themal conductivity, themal diffusivity and heat capacity pe unit volume, A BT and R ST. Estimated uncetainties in and ae about 5% and 7% espectively. ( T ) ( T ) whee A(W - m K) and B(W - m) ae constants elated to the scatteing popeties of phonons. A is elated to scatteing of phonons by impuities and impefections [0]; and B is elated to phonon-phonon scatteing and is appoximately popotional to an invese powe of sound velocity []. The physical justification of the tem A is the existence of numeous additional scatteing centes fo phonons in mateials, caused by stuctual and chemical impefections and the influence of the gain boundaies. Eq.2 is analogous to Matthiessen s ule of electical esistivity in metals, which is: ( ), (3) e eo e T whee eo is the extapolated electical esistivity at 0 K and is called the esidual esistivity. This is the tempeatue independent pat of the esistivity. That is, the value of eo depends upon the concentation of impuities and othe impefections in the sample. It can be taken as the measue of impuity of the specimen. The tempea-

4 tue dependent pat of esistivity e (T ) is decided by the phonons. It is called the lattice esistivity [2]. Fo diffusivity (having the same tend as themal conductivity), the analogous equation was assumed to be: R ST (4) T The esults obtained using Eqs.2 and 4 along with the expeimental esults / values of and ae shown in Figue 3. The coelation coefficients fo themal conductivity and themal diffusivity ae also listed in Table. The data appea to fit vey well (above 97%). The values of C p can be calculated using Eq.. Eq.2 may also be witten as [3,4]: ( T ) ( bt ) /( bt ), (5) whee (W m - K - ) is the themal conductivity at a efeence tempeatue and b(k - ) is a single tempeatue coefficient of themal conductivity paamete, contolling the tempeatue dependence of themal conductivity. b is elated to paametes A and B of Eq.2, as: b = B/A (6) The efeence value of themal conductivity ( ) is simply deived fom Eq.2, using T T (efeence/ oom tempeatue). It is to be noted that each of the Eq.2 o 5 is the modified Eucken s ule. A ecently poposed empiical model [6] fo the pediction of themal conductivity as a function of tempeatue is: mφ T, (7) e s f T whee m is the empiical coefficient whose value can be detemined at suitable tempeatues, using the coesponding expeimental values of themal conductivity and the oom tempeatue values of Φ, and s ; by A. Zeb et al. / Natual Science 2 (200) f m ft T / exp / s 629 (8) The empiical coefficients, exponents o adjustable paametes may vay accoding to the suite of ocks. Theefoe, the extapolations of empiical models to suites of ocks othe than those used in developing these models may not be satisfactoy [5]. 4. RESULTS AND DISCUSSION It is well known that themal tanspot popeties of poous ocks depend upon thei stuctue, mineal composition, poosity, density, the ability of thei constituent mineals to conduct heat, tempeatue, pessue; etc. The tempeatue dependence of themal conductivity and themal diffusivity in the tempeatue ange 83 to 483 K at suitable intevals is shown in Figue 3. It is obseved that themal conductivity deceases in the measued tempeatue ange. This is in ageement with the theoy of themal conductivity. The themal diffusivity shows deceasing tend; again in ageement with the theoy. Fo dunite samples, in Eq.5, the empiical coefficient b is taken to be which is equal to the mean of all the values of b listed in Table. Similaly the mean value of is found to be W m - K - at 303 K. Using these values of adjustable paametes and b, themal conductivity values ae pedicted along with the expeimental measuements and ae plotted in Figue 4. It is to be noted that in Eq.5, all the paametes effecting themal conductivity (such as poosity, density, themal conductivity values of constituent mineals and the fluid inside the poes, etc.) ae dumped into adjustable paametes and the eos in pedicting themal conductivity ae only up to 0% in the tempeatue ange 243 K to 333 K, which is the most inteesting ange fo constuction puposes. Table. Paametes A, B, R and S fo the themal esistivity ( T ) and the invese of themal diffusivity ( T ) of dunites as epesented by Eqs.2 and 4. and ae the espective coelation coefficients. Specimen A B R S b = B/A W - m K 0-4 W - m m -2 sec 0-3 m -2 sec K - K - - W m - K Dn Dn Dn Dn Dn Dn

5 630 A. Zeb et al. / Natual Science 2 (200) Figue 4. The compaison of expeimental and pedicted themal conductivity of dunite samples as a function of tempeatue. As fa as the esults of Eq.7 ae concened, these ae faily good at oom tempeatue and above (with in 0%) [3]. But, this poposal is not ecommended fo the pediction of themal conductivity in tempeatue ange lowe than the oom tempeatue. This is the limitation of this model. It is because of the fact that we have used the oom tempeatue values of s, f and Φ in Eqs.7 and 8. This makes the value of m (calculated by Eq.8) negative in the tempeatue ange lowe than oom tempeatue. This eveses the effect of second tem on R.H.S. of Eq.7. As a consequence, we get the inceasing effect in themal conductivity with tempeatue due to this tem which is against expeimental esults. 5. CONCLUSIONS To pedict the themal conductivity of dunite samples as a function of tempeatue, modified Eucken s ule has been used. It is noted that the expeimental themal conductivity and the pedicted conductivity ae in good ageement (up to 0%) in the tempeatue ange 243 K to 333 K, which is impotant fo constuction puposes. 6. ACKNOWLEDGEMENTS The authos wish to thank M. Zulqunain Ali, M. Matloob Hussain (Eath Science Depatment, QAU) and M. Muhammad Saleem Mughal (Statistics Depatment, QAU) fo helpful discussion. The autho (Auang Zeb) is gateful to Highe Education Commission, Pakistan fo suppoting the doctoal studies financially. REFERENCES [] Cengel, Y.A. and Tune, R.H. (200) Fundamentals of themal-fluid sciences. McGaw-Hill, Boston, 609. [2] Rzhevsky, V. and Novik, G. (97) The physics of ocks. Mi Publishes, Moscow. [3] Auangzeb, Mehmood, S. and Maqsood, A. (2008) Modeling of effective themal conductivity of dunite ocks as a function of tempeatue. Intenational Jounal of Themophysics, 29(4), [4] Gustafsson, S.E. (99) Tansient plane souce techniques fo themal conductivity and themal diffusivity measuements of solid mateials. Review of Scientific Instuments, 62(3), [5] Auangzeb, Ali, Z., Gumani, S.F. and Maqsood, A. (2006) Simultaneous measuement of themal conductivity, themal diffusivity and pediction of effective themal conductivity of poous consolidated igneous ocks at oom tempeatue. Jounal of Physics D: Applied Physics, 39(7), [6] Auangzeb, Khan, L.A. and Maqsood, A. (2007) Pediction of effective themal conductivity of poous consolidated media as a function of tempeatue: A test case of limestones. Jounal of Physics D: Applied Physics, 40(6), [7] Maqsood, M., Ashad, M., Zafaullah, M. and Maqsood, A. (996) Low-tempeatue themal conductivity measuement appaatus: Design assembly, calibeation and measuement on (Y23, Bi2223) supeconductos. Supeconducto Science and Technology, 9(4), [8] Maqsood, A. and Rehman, M.A. (2003) Measuement of themal tanspot popeties with an impoved tansient plane souce technique. Intenational Jounal of Themophysics, 24(3), [9] Maqsood, A., Rehman, M.A., Gumen, V. and Haq, A. (2000) Themal conductivity of ceamic fibes as a function of tempeatue and pess load. Jounal of Physics D: Applied Physics, 33(6), [0] Schatz, J.F. and Simmons, G. (972) Themal conductivity of eath mateials at high tempeatues. Jounal of Geophysical Reseach, 77(35), [] Ziman, J.M. (960) Electons and phonons. Oxfod Univesity Pess, London. [2] Vijaya, M.S. and Rangaajan, G. (2004) Mateials science. McGaw-Hill, New Delhi. [3] Kukkonen, I.T., Jokinen, J. and Seipold, U. (999) Tempeatue and pessue dependencies of themal tanspot popeties of ocks: Implications fo uncetainties in lithosphee models and new laboatoy measuements of high-gade ocks in the Cental Fennoscandian Shield. Suveys in Geophysics, 20(), [4] Jokinen, J. (2000) Uncetainty analysis and invesion of geothemal conductive models using andom simulation methods. Oulu Univesity, Oulu. [5] Someton, W.H. (992) Themal popeties and tempeatue elated behaviou of ock/fluid systems. Elsevie, New Yok.