Fractal analysis of thermal conduction of loose coal

leonlineatwww.sciencedirectcomMININGDirectSCIENCE ANDTECHNOLOGYELSEVIERMining Science and Technology 20(2010)0831-0834www.elsevier.com/locate/jcumtfractal analysis of thermal conduction of loose coalWANG Yijiang, ZHOU Guoqing", WEI Yazhi, ZHOU YangSchool of Mechanics Civil Engineering, China University of Mining Technology, Xuzhou 221116, ChinaState Key Laboratory of Geomechanics Deep Underground EngineeringChina University of Mining Technology, Xuzhou 221008, ChinaAbstract: For deep mining engineering, heat transfer of coal mass is a vital factor in the thermal environment of coal mines. Inorder to study the thermal conduction mechanism, we obtained gray images of coal mass microstructure by scanning samples with adigital microscope. With the use of Matlab, these gray images were transformed into binary images, which were then transformedinto a corresponding matrix consisting only of the values 0 and 1. According to the calculation method of box-counting dimension,we calculated the fractal dimension of the loose coal to be approximately 1. 86. The thermal conductivity expressions of loose coalwere derived based on the simulation method of thermal resistance. We calculated the thermal conductivity of loose coal by using afractal model and compared the calculated values with our experimental data. The results show that the test data show an encouraging agreement with the calculated values. Hence fractal theory is a feasible method for studying thermal conductivity of loose coal.Keywords: loose coal; thermal conductivity; fractal theory; box-counting dimension1 Introductiontypical in the use of fractal theory to study the ther-mal mechanism of random media 5-b. As is wellGiven the long-term and large-scale exploitation of known, coal mass is a type of porous medium, somineral resources, high-grade mineral resources bur- fractal theory can also be used to study thermal con-ied in shallow seams are being gradually exhausted, duction of loose coal.so more and more coal mines are faced with the chal-lenge of deep mining since the middle of the 20 2 Fractal study of loose coalcentury. The most significant differences betweendeepg and shallow mining are the special cir- 2.1 Fractal theorycumstances of rock mass", including high groundEuclidean geometry only applies to describingstress, high geothermal and high hydraulic pressure. bodies with integer dimensions, where e.g., the diFor heat harmful in coal mines, heat transfer of coal mension of length, area and volume is respectively 1mass (or rock mass)is the most important factor in 2 and 3, but is not suitable for porous media such assoil and loose coal. The fractal geometry mainly fo-the heat transfer mechanism of rock or coal masses Is cuses on the irregular and complex characteristics ofurgently required for thof thermal disastersCurrent research on thermal conductivity in coal is ture and investigates the internal rules of nonlinearcomplex systems, i.e., the dimensions in fractal ge-largely experimental. The mean value of many ometry are continuous. A fractal dimension is a pa-experimental results is regarded as the thermal conrameter describing the complexity and irregular exductivity of coal mass, but this requires conducting a tent of fractal material. The mathematical expressionlarge number of experiments and cannot elucidate the of a fractal can be expressed asISIheat transmission process and thermal conductionmechanism. Since Mandelbrot put forward his fractalN∝6theory, researchers began to use it to describe the heat where N is the feature product of a fractal body(suchtransmission process in porous media and Thovert iske measurement scale中国煤化工 lidan space,NisReceived 12 February 2010, accepted 16 May 2010CNMHgme while d=l, 2.Corresponding author. Tel: 86 516 83995078E-mailaddresswyj_cumt@163.comFractal theory is widely used to conduct an inves-doi:10.016s16745264(09×02908tigation of thermo-physical properties of porous meMining Science and TechnologyVoL20 No 6we rarely pay attention to the thermal conduction ent sized particles. 2)There are many connected r.dia. Current studies have mainly focused on soil and 1)Loose coal is composed of coal mass of diffemechanism of loose coal"-1o. with the help of mi- unconnected cracks in loose coal, which is muchcro-structural experiments, we studied the thermal clearer than that of intact coal. 3)The distribution ofconductivity mechanism of loose coal, using fractal cracks cannot be described in Euclidean space. Morespecifically, loose coal is composed of the coal body2.2 Micro-structural experimentand fluids (e.g air and water)in the cracks. Duringheat transfer, the distribution of cracks and their fluidThe intact coal for testing was collected from a characteristics have a considerable effect on thecoal mine. The coal mass was scanned by using a thermo-physical properties of loose coal.KH3000 digital microscope in the State Key Labora-tory for GDUE. As shown in Fig. 1, rough2.3 Calculation of fractal dimensionscracks are much in evidence on the surface of the coalThe current fractal dimensions contain both Hausmassdorff dimension and box-counting dimension. Be-The distribution of cracks was studied in previ- cause the lower bound of the Hausdorff dimension isously, using fractal measuring method of trace extremely difficult in theoretical calculations, boxlength number of cracks and fractal measuring counting dimension is widely used. The definition ofmethod of the number of grid scale-visual cracks. The a box-counting dimension", F, is assumed to be afractal dimensions of two measuring methods show nonempty bounded subset of R and No(F) is thethat coal is a fractal porous medium with a large smallest number of sets covering F, with the largestnumber of cracks, which affects the permeability of diameter of 8. If the lower bound of the box-countingoal seams, such as gas diffusion and water seepage. dimension is equal to its upper bound, the boxcounting dimension of F can be expressed as Eq- (2)dim.F= lim log N3(F)porous medium. For example, Chen and Shi proposeda method of particle area of a section plane, equal tothe definition of a box-counting dimension. Theessence of method is the calculation of the averagelid area S around one particle, with a differentFig. I Image of compact coal(200 timeslength, X. The relation is shown as Eq- 3)nS′=lnc+dlnXAlthough coal is a type of porous medium, itswhere c is a constant and d a fractal dimension. Eq (3)stant,with changes in atmospheric temperature and provides a new method to calculate the box-countingsure, We considered the thermal conductividimension of porous media. However, there are manyproblems for calculating the average solSolid coal to be constant. Loose coal samples, with Especially for a real random medium, it is difficult todiameters of 0.6-1. 2 mm, were randomly aclated to simulate an actual situation, illustrated in Fig. calculate the average solid areaBased on image storage theory, Peng et al. ad-vanced a pixel matrix analysis for calculating fractaldimensions[ 31. It is a much more feasible methodthan that provided by Eq (3)Scan pictures are transformed into a binary imagewith only white and black colors for every pixel, asshown in Fig 3. The binary images arformed, by Matlab, into a matrix which consists onlyof the values 0 and 1. If the color of the pixel is white,the corresponding matrix element is 0, otherwise theelement is 1The matrix with values 0 and 1. is divided intoFig. 2 Image of loose coal (50 timesman中国煤化工hmnAs can be seen, the white and gray areas are thesolid coal and the dark areas the cracks. Some fea- N(n)wnieCNMHGg, is replaced bynumber oI nxn square matricestures of loose coal are as followsconsisting value 1. The limit 8-0 can be obtainedfractal analysis of thermal conduction of loose coalfrom the different orders of the nxn square matrices,so that the box-counting dimension can be expressedEq1(4)dimF=lirlog N(n)BBased on this equation, the pixel matrix is calcuted, with the results shown in Fig. 4. The fractaldimension of loose coal is about 1.86Fig 5 Thermal conduction of coal mass cellFig 6 Thermal resistance of a single coal mass cellThe entire thermal resistance of the coal mass isFig 3 Binary image(h+2m)aa is the cell thermal conductivity, W/(mK)rding to the series and parallel laws of thermalresistance. the entire thermal resistance can also beexpressed asFig 4 Calculation of box-counting dimensionr42n+2Substituting Eqs. (5)and (7)into Eq (6), we can3 Analysis of thermal conductionobtain the thermal conductivity d of coal mass cell3.1 Fractal study of thermal conductivityx=2m(-2m)1+(M+4m2(8For a single loose coal cell, the section can be sim-(-2m)h+2m)x+2m(h+2mplified as a rectangle, composed of solid coal and where p is the void ratio of a loose coal section;s thethermal convection and thermal radiation of coal and three parameer&.e coal cell area; the relation of thecracks can be neglected. If we assume that there is nothermal contact resistance between the cracks andsolid coal, then the heat transfer process is only ther-s=2m(h+Dmal conduction. We set the length of the thermaltransfer cell as L. coal thickness as h. the crack thickS=l(h+2m)cording to the simulation analysis of thermal resis- the relation of s and S is expressed by Eq 10/ asion,ness as m and thermal flow direction horizontal. ac-tance, we obtained a thermal resistance relationshipas shown in Fig. 6. Additionally, we set the thermas=CSconductivity of the crack fluid as Ag, the thermalWe calculated the value of C=0. 28. According toconductivity of the solid coal medium as a, and as- the definition of a fractal dimension, the void ratiosumed the thermal flow as homogenous in all parts. should meet Eq (11):Based on this hypothesis, theach part is:p=CS(11)=r=m(M2)twee中国煤化工void ratio can be2=(-2m)(h)CNMHGe diameter ofloose CuauU1,∠nm, with an aver4=5=l(m1)age of 0.9 mm. In calculation, we assumed that theMining Science and TechnologyVol,20 No 6diameter of the loose coal was 0.9 mm, and both than the test value. In spite of that, it is feasible tolength and width of a coal cell were 1.1 mm. Given study the thermal conduction mechanism by fractalEqs. (9)and(10), we calculated =0.33If we assume that there is no water in the loosecoal sample, the crack is only filled with air. At at- 4 Conclusionsmospheric temperature and pressure, the thermalconductivity of air is 0.0256 W/(mK), while the1)We studied the thermal conduction mechanismthermal conductivity of coal is 1.45 W/(m-K). Sub- of loose coal by fractal theory and calculated thestituting Eqs. (9), (10)and into Eq (8), we obtained thermal conductivity. The results show that the calcuthe element of thermal conductivity i=0. 10 W/(m-K). lated values coincide with the test data. fractal theoryAccording to the self-similarity of fractal theory, the is a new promising method for studying thermalthermal conductivity of loose coal is 0.10 W/(mK). conduction of porous media such as loose coal. It is3.2 Thermal conductivity testalso suitable for research in heat transfer of coal massin mines where thermal heat is dangerous becauseThe thermal conductivity of our loose coal samples residual coal and coal gangue can be self-ignitingwas tested by the Quick-line thermal property ana-2)We extended the method of calculating box-yzer, as shown in Fig. 7.counting dimensions. Binary images were transferredby Matlab into a corresponding matrix only with 0and 1 as values. We calculated the fractal dimensionof loose coal to be approximately 1.863)Loose coal is composed of solid coal and cracksThermal conductivity is associated not only with thedistribution of cracks but also with the thermo-hysical characteristics of fluid in cracksAcknowledgementsFig 7 Quick -line testerFinancial support for this study, provided by thetest results are in the rangehe range of 0. 1l to 0. 12 National Natural Science Foundation of ChinaW/(m-K), as shown in Table 1, which coincide with (Nos. 50534040 and 50974117)and the Researchthe calculated valFund of the State Key Laboratory of Coal ResourcesThere are two factors affecting thermal conductiv-&Mine Safety, CUMT(No. 07KF10), is gratefullyity of coal mass1)Porosity: the relation between porosity and acknowledged.thermal conductivity can be expressed by ArchiesReferenceslaw. It is an empirical correlation which is mostlused for geological material and can be expressed as: [U He M C, Xie H P, Peng S P, Jiang Y D Study on rock(12)mechanics in deep mining engineering. Chinese Journalof Rock Mechanics and Engineering, 2005, 24(1)where p is porosity and m an exponent, usually rang-803-2813. (In Chinese)ng from 1.3 to 2.5. From Eq (12)we see that th[2] Li J W, Ge L M, Xu J C, Wen H Measurement of ther-mal conductivity in loose coal bulk Joumal of Liaoningthermal conductivity mostly depends on the porosityTechnical University, 2004, 23(1): 5-8.(In Chinese)of the coal mass[3] Chen Q H, Zhang G S, Liang H Z, Xu LJ. Design on2)Moisture: as is well known, thermal conductiv-thermal conductive coefficient measuring system of bulkof water is twenty times than that of air. Moisturecoal mass. 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