A general solution and approximation for the diffusion of gas in a spherical coal sample

International Journal of Mining Science and Technology 24(2014)345-348Contents lists available at Science DirectInternational Journal of Mining Science and TechnologyELSEVIERurnalhomepagewww.elsevier.com/locate/ijmstA general solution and approximation for the diffusion of gasn a spherical coal sampleWang Yucang Xue Sheng, Xie JunCSIRO Earth Science and Resource Engineering, Kenmore, QLD 4069, AustraliaARTICLE INFOABSTRACTArticle historThe square root relationship of gas release in the early stage of desorption is widely used to provide aReceived 10 October 201imple and fast estimation of the lost gas in coal mines. However, questions arise as to how the relationReceived in revised form 15 November 2013Accepted 8 December 2013ship was theoretically derived, what are the assumptions and applicable conditions and how large theAvailable online 29 April 2014error will be. In this paper the analytical solutions of gas concentration and fractional gas loss for the dif-fusion of gas in a spherical coal sample were given with detailed mathematical derivations based on thediffusion equation. The analytical solutions were approximated in case of small values of time and theCoal contenterror analyses associated with the approximation were also undertaken. The results indicate that thesquare root relationship of gas release is the first term of the approximation, and care must be takenSpherical coal samplein using the square root relationship as a significant error might be introduced with increase in the losGas diffutime and decrease in effective diameter of a spherical coal sampleApproximationo 2014 Published by Elsevier B V on behalf of China University of Mining technologError analysis1 Introductionsignificantly dependent on a number of factors such as samplshape, retrieval time, and physical character of the sample [7, 8Gas content of coal is widely used in gas emission estimation, Crank did present in his book the analytical solutions of diffusionand gas outburst proneness assessment in underground coal min- from cylindrical and spherical samples, but still no mathematicalng [1 The commonly used method for the estimation of the gas derivations were given [ 9]. This raises the questions: how the relacontent is the direct desorption method. This method is based on tionship was theoretically derived? What are the assumptions andobservations of gas release from newly obtained samples, and typ- applicable conditions and how large the error will be? To answerally involves extracting a coal sample, enclosing it in a sealed some of the questions and improve the accuracy of estimation, thiscontainer and measuring the volume of gas released. With the paper gives detailed derivations of a general mathematical solutiondirect method the total gas content of a coal sample is made of for the diffusion of gas in a spherical coal sample and an approxithree parts: lost gas, measurable gas, and residual gas [2-5]. The mation solution. The detailed derivations of a general mathematilost gas(Q1)is the gas desorbed from the sample before it is placed cal solution for the diffusion of gas in a cylindrical coal sample andin the canister. The measurable gas(Q2)is the gas desorbed during an approximation solution have been published by the same grouptransport and in the laboratory. the residual gas( Q3)is the gas still of authors of this paper [10]. These mathematical derivations alsocontained in coal at one atmospheric pressure. While Q2 and Q3 have particular relevance in determining the sorption time and dif-an be directly measured, Q1 has to be estimatedfusion coefficient in our numerical models of coal and gas outThe"lost gas"estimation method was firstly described by Ber- bursts (11, 12tard et al. [6]. It was stated in the paper that early in the desorptionprocess the volume of gas released from coal was proportional tothe square root of time. However no details were given as hethe relationship was theoretically derived except mentioning that 2. Diffusion equation and the initial and boundary conditionsit was based on kinetics of gas desorption from coal. Since then thisGas release from coal is considered to be the process of diffusionsquare root relationship has been widely used as a standard lost in the coal matrix and desorption at surface. The diffusion throughgas estimation method. The relationship has been found to bethe matrix is asoncentration gradient-driven andusually modeledond law of Diffusion [13]. For aCorresponding author. Tel. +61 7 33274531spherical sample中国煤化工ient(D), the gasE-mail address: Yucang. wang@csiro. au(YconcentrationCNMCOordinatehttp://dx.doiorg/10.1016/j.ijmst.2014.03.0102095-2686 2014 Published by Elsevier B V on behalf of China University of Mining TechnologyY, Wang et al/Internal of Mining Science and Technology 24(2014)345-348sphere (r)only then the diffusion equation derived from Ficks Sec- 3. 2. Solution of the fractional lossond Law of Diffusion is given asLet the total gas volume in the sample at time t be Q, we then0≤r>0haver orThe initial and boundary conditions can be expressed as:QC. 4r2 dr0,0≤r0(4) At t-O. the initial total concentration is Qo=4r@ Co. the totalwhere a is the radius of the sphere Co the initial uniform concentra- desorbed gas M until time t istion;and C, the constant concentration at the surface of the sphere. Mr=Qo-QLet u=Cr, Eqs. (1)-(4)becomeDn2T2tat art=0,0lerfc (2n+1ja-r-erfc(2n+1)a.Fig 3. The relative error of the approximationTotal concentration at time t is6. Error analysescC.4丌r2d=4a3C0+a(C1-C0)47(T1-T2)To study the error introduced by the approximation, the fullsolution represented in Eq. (17)and approximation representedwhere T,=Jo rEn-oerfci2npa-dr and T2=Jo rEn-oerfci2njDatrin Eq (22 )are computed and the ratio of gas diffused to gasable to be diffused is plotted against Dt/a' in Fig. 2. In fig. 2, theBy applying the Laplace transform of partial function to T1 and solid line is the full solution and the other three dotted lines repre-sent three approximations taking the first term only the first term4丌a3and the third term without the error function terms and the firstQCo+a(C1-Co).4πterm and the third term plus the error function term with nin Eq (22 )respectively. It can be seen from Fig. 2 that the deviation2vD∑2 a.ierfc-2avD_Dtof these approximations from the full solution is significant andincreases in general with the increase of Dt/a. Fig. 3 plots theAt t=0, Q0=4r Co At time t, the total desorbed gas is calculatedelative errors introduced by the approximations. It is clear fromthe approximation with only the first term, whichcorresponds to the square root of time plot used to assess the lostMt=Q0-Qgas in core desorption, has the most deviation from the full solu--a(C1-C0):4tion or relative error and the error increases with the increase ofDt/a2. The error can exceed 10% if the value of dt/a' is more than√Dt0.0122D>2aier、However. M、 cannot be calculated using M、=lim、M, since eq.. Conclusions(21)is not valid when t -oc. Fortunately by physical intuition it isclear that Eq. (16)still holds. ThereforeThe diffusion of gas from a spherical sample is studied in detailFour major mathematical formulas regarding the concentrationDt+2>ierfcand the ratio of desorbed gas after time t over the total gas are2) derived. Two of them(Eqs. (15)and (17))are the analytical solutions and the othere their respectivewhere ierfc(x)= ierfc()di(n=1, 2, .. )are the iterated inte- small time appro中国煤化工 -elationship of gasgrals of the complementary error function. It can be easily seen that release in the earlCNMH Gund to be the firstthe square root relationship of gas release in the early stage of gas term of the approxindliunlmn ty. l4 uy igllunng the error funcdesorption is just the first term of the approximation in Eq. (22) tion term and the linear term The analytical solution(Eq. (17))by ignoring the error function term and the third linear termconsists of infinite terms, and converges fast for large values ofY. Wang et al. /Internal of Mining Science and Technology 24(2014)345-348Dt a2, but for small values of Dt/a2, the number of terms required toReferencesobtain the certain precision dramatically increases. Therefore, inpractical applications, the small time approximation of Eq( 22), 1L, Xie Wang YC. Status and prospects of gas content basedespecially the first vt term, is used to provide a simpler and fasteritburst control technology in Huainan. Int J Min Sci Technol, in pressestimation of the lost galatzel S]. Measuring the gas content of coal: a review. Int JCoal geol1998;35:311-31Cares must be taken in the lost gas estimate with the conver[3] Liang B, Sun W], Qi QX, Li HY. 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In: Proceedings of the internation.10% the lost time should be less than 20 min If sample is well frac-ind control of high gasns and outbursts inunderground coal mines, Wollongong, Australia, 1995tured (i.e. a smaller effective diameter) then the lost time must be[8 Mavor MJ. Pratt T]. Improved methodology for determining total gas content.much shorter than 20 min to keep the error under 10% In general ashorter sampling time, a larger sample size and smaller diffusionof lost gas models, Vol. Il, Gas Research Institute, Report GRI-94/0429, 1996.9] Crank J. The mathematics of diffusion. Oxford: Oxford University Press: 1975coefficient will introduce less error in the lost gas estimation[10] Li YB, Xue S, Wang JF, Wang YC, Xie J Gas diffusion in a cylindrical coal sampleIt should be noted that the general solutions and approxima-a general solution, approximation and error analyses. 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Oxford: Oxford Universitybe dependent upon temperature, pressure and current gas content,the number and sizes of fractures, and the distribution of pore sizes 151 Lama RD, Bodziony J Outburst of gas, coal and rock in underground coal[15]. In such cases, cares should be taken when using Eqs. (15).(17),(20),and(22)中国煤化工CNMHG