Numerical modeling and experimental validation of fractional heat transfer induced by gas adsorption in heterogeneous coal matrix

Abstract Despite one fundamental issue in the adsorption theory of coalbed methane, little is known about the thermodynamic properties of gas adsorption in a porous coal matrix. In this work, considering the heterogeneity of pore structure and the exothermic characteristics of gas adsorption, a fractional heat conduction model with an unsteady volumetric heat source is proposed to study the heat transfer process induced by gas adsorption in a heterogeneous coal matrix. The heat conduction equation with a fractional time derivative is discretized by using an implicit numerical method based on the generalization of a standard finite-difference scheme. First, to validate the fractional heat conduction model, gas adsorption experiments on a microcalorimeter were carried out on 5 g coal samples of 0.3 mm diameter at 25 °C. The experimental heat flux with initial adsorption pressures of 3.23 bar, 5.83 bar and 9.77 bar increases rapidly from zero to peak values of 7.17 mW, 12.05 mW and 16.81 mW in less than 7 min (i.e., fast thermal diffusion stage) and then decreases slowly to zero again in approximately 2 h (i.e., slow thermal diffusion stage). It is revealed that for all tested gas pressures the fractional heat conduction model with a fractional order α = 0.86 can reproduce the experimental process of heat flux with better accuracy than the Fourier law-based model (i.e., α = 1 ), suggesting that anomalous thermal diffusion is the governing heat transfer process of gas adsorption in the coal matrix. Second, the spatial distribution and temporal evolution of temperature patterns with different model parameters are numerically simulated. It is found that the time to reach the peak temperature decreases from 760 s at the center of the coal particles to 490 s at the boundary. Finally, the parametric sensitivity of the thermodynamic properties of gas adsorption such as temperature, heat flux and integral adsorption heat is discussed in detail. Particularly, it is shown that as one of the most important thermodynamic parameters, the integral heat is very sensitive to the fractional order α . In the case of 3.23 bar, if α increases from0.75 to 1, while other model parameters remain unchanged, the integral heat could be enhanced from 1.1 J/g to 8.5 J/g.

[1]  Sevket Durucan,et al.  A bidisperse pore diffusion model for methane displacement desorption in coal by CO2 injection , 2003 .

[2]  I. Podlubny,et al.  Modelling heat transfer in heterogeneous media using fractional calculus , 2013, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[3]  A. Busch,et al.  CBM and CO2-ECBM related sorption processes in coal: A review , 2011 .

[4]  Dong Zhou,et al.  Distribution law of temperature changes during methane adsorption and desorption in coal using infrared thermography technology , 2018 .

[5]  I. Gray,et al.  Reservoir Engineering in Coal Seams: Part 1-The Physical Process of Gas Storage and Movement in Coal Seams , 1987 .

[6]  F. Zhou,et al.  An anomalous subdiffusion model with fractional derivatives for methane desorption in heterogeneous coal matrix , 2015 .

[7]  Chang-Ha Lee,et al.  Adsorption characteristics of CO2 and CH4 on dry and wet coal from subcritical to supercritical conditions , 2011 .

[8]  M. Bülow,et al.  Measurement of sorption equilibria under isosteric conditions The principles, advantages and limitations , 2002 .

[9]  Jishan Liu,et al.  Experimental study of impact of anisotropy and heterogeneity on gas flow in coal. Part II: Permeability , 2018, Fuel.

[10]  F. Zhou,et al.  Effects of Coal Functional Groups on Adsorption Microheat of Coal Bed Methane , 2015 .

[11]  Y. Povstenko.  FRACTIONAL HEAT CONDUCTION EQUATION AND ASSOCIATED THERMAL STRESS , 2004 .

[12]  Jianhong Kang,et al.  A fractional non-linear creep model for coal considering damage effect and experimental validation , 2015 .

[13]  Fawang Liu,et al.  An improved heat conduction model with Riesz fractional Cattaneo-Christov flux , 2016 .

[14]  Xiu Yang,et al.  Numerical analysis for electroosmotic flow of fractional Maxwell fluids , 2018, Appl. Math. Lett..

[15]  Changpin Li,et al.  Fractional differential models for anomalous diffusion , 2010 .

[16]  Xiaoyun Jiang,et al.  Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation , 2011 .

[17]  L. Connell,et al.  Modelling permeability for coal reservoirs: A review of analytical models and testing data , 2012 .

[18]  Feng Gao,et al.  Evolution of coal self-heating processes in longwall gob areas , 2015 .

[19]  D. W. Pollock,et al.  Gas transport in unsaturated porous media: The adequacy of Fick's law , 1989 .

[20]  Yu Huaizhong,et al.  A thermal non-equilibrium model for 3D double diffusive convection of power-law fluids with chemical reaction in the porous medium , 2017 .

[21]  Greg Duffy,et al.  Effect of coal properties on CO2 sorption capacity under supercritical conditions , 2008 .

[22]  B. Henry,et al.  The accuracy and stability of an implicit solution method for the fractional diffusion equation , 2005 .

[23]  Xiaojie Guo,et al.  The influence factors for gas adsorption with different ranks of coals , 2018 .

[24]  Zhaofeng Wang,et al.  Physical Simulation of Temperature Influence on Methane Sorption and Kinetics in Coal (II): Temperature Evolvement during Methane Adsorption in Coal Measurement and Modeling , 2015 .

[25]  Satya Harpalani,et al.  Gas diffusion behavior of coal and its impact on production from coalbed methane reservoirs , 2011 .

[26]  Debadutta Mohanty,et al.  Thermodynamics, kinetics and modeling of sorption behaviour of coalbed methane – A review , 2016 .

[27]  F. Gómez,et al.  Fractional thermal diffusion and the heat equation , 2015 .

[28]  Z. Weishauptová,et al.  The effect of moisture on the sorption process of CO2 on coal , 2012 .

[29]  Xuetao Cheng,et al.  THE DIFFUSION MODEL OF FRACTAL HEAT AND MASS TRANSFER IN FLUIDIZED BED A Local Fractional Arbitrary Euler-Lagrange Formula , 2015 .

[30]  Mingyu Xu,et al.  The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems , 2010 .

[31]  Frank L. Williams,et al.  Diffusion models for gas production from coals: Application to methane content determination , 1984 .

[32]  Y. Niibori,et al.  Fractional diffusion modeling of heat transfer in porous and fractured media , 2016 .

[33]  Yanbin Yao,et al.  Pore structure and its impact on CH4 adsorption capacity and flow capability of bituminous and subbituminous coals from Northeast China , 2013 .

[34]  Tongqiang Xia,et al.  Numerical modeling and experimental validation of anomalous time and space subdiffusion for gas transport in porous coal matrix , 2016 .

[35]  Norbert Skoczylas,et al.  Sorption Rate of Carbon Dioxide on Coal , 2013, Transport in Porous Media.

[36]  Yuanping Cheng,et al.  Modeling and experiments for transient diffusion coefficients in the desorption of methane through coal powders , 2017 .

[37]  Haitao Qi,et al.  Transient fractional heat conduction with generalized Cattaneo model , 2014 .

[38]  Y. Niibori,et al.  Analysis of Water Injection in Fractured Reservoirs Using a Fractional-Derivative-Based Mass and Heat Transfer Model , 2014, Mathematical Geosciences.

[39]  Yangquan Chen,et al.  Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion , 2011, Comput. Math. Appl..

[40]  Suzanne D. Golding,et al.  Numerical simulation of multicomponent gas diffusion and flow in coals for CO2 enhanced coalbed methane recovery , 2007 .