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Numerical study of a domain decomposition method for a two-scale linear transport equation

  • Received: 01 September 2005 Revised: 01 November 2005
  • 35B25, 65M60, 76R50, 82C70.

  • We perform a numerical study on a domain decomposition method proposed in [13] for the linear transport equation between a diffusive and a non-diffusive region. This method avoids iterating the diffusion and transport solutions as in a typical domain decomposition method. Our numerical results, in both one and two space dimensions, confirm the theoretical analysis of [13]. We also provide an improved second order method that provides a more accurate numerical solution than that proposed in [13].

    Citation: Xu Yang, François Golse, Zhongyi Huang, Shi Jin. Numerical study of a domain decomposition method for a two-scale linear transport equation[J]. Networks and Heterogeneous Media, 2006, 1(1): 143-166. doi: 10.3934/nhm.2006.1.143

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  • We perform a numerical study on a domain decomposition method proposed in [13] for the linear transport equation between a diffusive and a non-diffusive region. This method avoids iterating the diffusion and transport solutions as in a typical domain decomposition method. Our numerical results, in both one and two space dimensions, confirm the theoretical analysis of [13]. We also provide an improved second order method that provides a more accurate numerical solution than that proposed in [13].


    1. Introduction

    Coal oxidation at ambient temperatures is considered the main reason for products quality and energy losses. Chemisorption and oxidation occur when coal is exposed to air at relatively low temperatures (below 100 °C). Moreover, as such processes are exothermal, this may also lead to self-heating (accelerating the oxidation reaction) and spontaneous combustion events during production, storage, transportation and utilization of coal [1,2,3].

    There exist a certain amount of instrumental methods and techniques for characterization of coals propensity to self-heating at oxidation. The most widely used one is the so-called R70 method [4,5]. The essence of this method is as follows. The grinded (with particle sizes of <212 µm) coal sample is dried in adiabatic oven at temperature of 105-110 °C for 15 hours under inert gas flow, then cooled to 40 °C in the same environment, and after that is being stored in the oxygen-rich air flow at the temperature of 40 °C. Under the latter conditions, coal oxidation is being initiated and the processes of self-heating are observed. The average rate of the coal heating from 40 to 70 °C is considered to be the index R70 (°C/h). The higher the value of such index, the more prone the coal is to spontaneous combustion. Such method is the most efficient one with respect to simple characterization of coals propensity to oxidation and self-heating, but unfortunately there exist some disadvantages connected with the method of samples preparation, namely, their drying at high temperatures which may lead to irreversible alterations of coals properties.

    Crossing Point Temperature (CPT) is another method widely applied for characterization of coals propensity to spontaneous combustion [6,7]. Here, the coal sample is placed in the oven which is being heated with constant rate. Within this method, a rate of coal sample heating is compared with the rate of the oven walls heating at the condition of constant air supply to the sample. In comparison with the aforementioned R70 method, CPT allows in situ coals samples testing, but still there are some disadvantages connected with the samples moisture, particles sizes, etc.

    Other promising methods of characterization of coals spontaneous combustion due to oxidation are thermogravimetric analysis (TGA) [8] and differential scanning calorimetry [9]. Despite the fact that these approaches also allow studying of coal in situ, they also have disadvantages, namely, the oxidation effects are observed here at the temperatures much higher than ambient (over 200 °C).

    In order to completely describe the behavior of systems prone to spontaneous combustion, one should have sufficient information on the mechanism and kinetics of the characteristic chemical reactions considering also temperature and the conditions of heat loss to the surroundings. The activity of processes in the system depends on numerous factors, including the chemical composition, supramolecular structure, and porosity of the coal, as well as ambient conditions (air humidity, air flow, etc.). Many authors report experimental data on the effects of individual factors on the chemical activity of coal and use them as the basis for kinetic models of self-heating (e.g. see [1,2,10,11,12,13] for review). Although, it seems difficult to develop a complete kinetic model allowing the behavior of coal under real conditions to be predicted.

    In order to simplify the description of the self-heating process, the interaction of coal with air is assumed to be a single exothermal reaction. The chemical activity of coal is estimated from apparent kinetic constants of the reaction, namely, the activation energy and the pre-exponential factor. They are commonly estimated using thermal methods [11,12,13,14,15,16,17,18] based on the results of direct laboratory measurements of thermal effects during oxidation. Kinetic constants evaluation within the thermal steady-state method is based on Frank-Kamenetskii (F-K) analysis [19] which was originally developed to determine the critical conditions of spontaneous combustion when the kinetics of the process is known. Thermal methods solve the inverse problem, i.e., determination of the kinetic constants when the parameters corresponding to the critical conditions of the system are known. This requires estimation of the relationship between the characteristic temperature and the critical F-K parameter (δсr) determining the critical conditions under which a steady-state temperature profile in the system becomes impossible. The parameter δсr is preliminarily calculated for the laboratory reactor used in the study; the characteristic temperature is estimated from the results of a thermal experiment on coal heating in the reactor with the temperature on its surface maintained constant.

    One of the most useful methods for evaluation of the relationship between δсr and the critical ambient temperature (Ts cr) (at which the temperature at the center of the sample drastically rises) is the traditional steady-state thermal method [3]. The kinetic constants are determined from the plot of versus 1/Tscr (where R is the characteristic size of the reactor) based on experimental data obtained for different sizes of laboratory reactors with the δсr parameter known for each of them. In thermal experiments, the crushed coal in a cylindrical laboratory reactor is placed into an oven where the constant temperature is maintained. The experiments are usually carried out at temperatures above 100 °C, because rapid heat loss to the surroundings of the sample makes it impossible to observe sample heating caused by the exothermal reaction at lower temperatures. It takes a series of thermal experiments to determine the critical temperature for each reactor.

    In order to reduce the number of experiments, a transient method (the Chen method) has been proposed [17,18]. The estimation of the kinetic constants in this method is based on determining the rate of temperature change at the center of the reactor at the moment of time when the thermal conduction term of the equation of heat balance near the center becomes zero. The temperature corresponding to this moment is estimated experimentally from the condition of equality of the temperatures at the center and a point near it; it is termed the crossing point temperature. This method is described in detail as applied to cylindrical reactors in the study [17]. In this study, the estimation of the crossing point temperature was based on determining the moment when only the radial component of the thermal conduction term became zero. Strictly speaking, however, the moment when the sum of the radial and vertical components of the thermal conduction term is equal to zero should be used in the calculations. As further development of the Chen method, a procedure has been suggested for estimating the kinetic parameters from the dependence of the dimensionless crossing point temperature (θcpt) on the F-K parameter [19]. This dependence for an infinite slab has been shown to have the form θcpt = 0.1δ.

    These transient methods for estimation of the kinetic parameters have allowed the number of experiments to be reduced compared to the steady-state method. The aim of the current work is to present a possibility of the same reduction in the context of the steady-state approach using experiments where critical conditions are not reached. To this end, a modified method for estimation of the kinetic parameters from the temperature at the center of a laboratory reactor in the steady-state mode is proposed. It was also established how the calculated values of the kinetic constants depend on whether or not the oxygen consumption in the oxidation reaction is taken into account.


    2. Modified Steady-State Analysis

    As noted above, the traditional steady-state approach to the estimation of the kinetic constants requires that the critical conditions characterized by the critical F-K parameter (δсr) be found. These conditions for each reactor are determined in a series of experiments at different temperatures on the sample surface. Let us consider how the results of a thermal experiment where the system goes to a steady state without reaching the critical conditions can be used for determining the kinetic constants. It is suggested that the relationship between the F-K parameter and the dimensionless temperature (θс) is used in the steady-state mode at the center of the laboratory reactor, where heating is the most intense.

    Coal heating is described by an unsteady-state energy balance equation for the case of an internal heat source:

    ρCpTt=kΔT+QW(T), (1)

    where T is the temperature (K), ρ is the bulk density of coal (kg m−3), Cp specific heat capacity of coal (J kg−1 K−1), k is the thermal conductivity of coal (W m−1 K−1), Q is the heat of oxidation (J kg−1) released as a result of the reaction. It is assumed that the reaction rate depends on temperature according to the Arrhenius law: W(T) = A exp(-E/RgT), where A is the pre-exponential factor (s−1) which includes the concentration of the reagents and E is the apparent activation energy (J mol−1), Rg is the universal gas constant (8.314 J mol−1 K−1). The consumption of the reagents in the course of oxidation is not taken into account, because this would substantially complicate the task. The use of Eq. (1) requires preliminary determination of parameters of the coal, including its density, thermal conductivity, and heat of reaction. In analyzing the results of thermal experiments, it is convenient to go from Eq. (1) to the equation describing the rate of temperature change:

    Tt=aΔT+Bexp(E/RgT). (2)

    Here a is the temperature conductivity coefficient (m2/s), B = QA/(Cpρ) is a parameter that includes the constant characterizing the reaction rate and the thermophysical constants of the system (Ks−1); B determines the rate of temperature rise as a result of the exothermal oxidation reaction under adiabatic conditions. Equation (2) allows calculation of the kinetic parameters E and B provided that the coefficient a is known. The number of variables in Eq. (2) can be reduced using the F-K approximation to the Arrhenius term [19]. In order to obtain scaled dimensionless variables the dimensional variables are divided by their scales: length, ξ = x/R, where R is the characteristic size (the radius of the reactor); time, τ = at/R2; and temperature, θ=E(TTs)/RgTs2, where Ts is the temperature of the reactor walls. Equation (2) in dimensionless variables takes the form

    θt=Δξθ+δexp(θ) (3)

    Here, Δξ is the Laplacian operator written in dimensionless variables and δ=ER2BRgTs2exp(ERgTs) is the dimensionless F-K parameter. The solutions of Eq. (3) are determined by the F-K parameter. The conditions of combustion are characterized by the critical value δсr. The condition δ = δсr corresponds to the critical state where steady-state distribution becomes impossible. At δ < δсr, a steady-state temperature distribution is formed in the system; at δ > δсr, heat generation exceeds heat loss to the surroundings, which leads to self-heating of the system followed by spontaneous combustion. The parameter δсr is calculated for a given system with known thermophysical parameters and a prespecified geometry.

    Following F-K method it is possible to obtain the correlation between parameter and a dimensionless temperature θс in a steady state inside the central point of the laboratory reactor. To do this it is suffice to solve the steady state temperature distribution problem taking into account the chemical reaction:

    Δξθ=δeθ (4)

    The solution of the equation (4) corresponds to a steady state temperature distribution. Numerical solution of Eq. (4) has yielded the values of the dimensionless temperature at the center of the reactor (θc) at different values of δ for a cylindrical reactor with a height-to-diameter ratio of h/d = 1 (Table 1). The boundary condition at the sample surface is θc = 0. It is convenient to represent the solution of this equation in a plot as the dependence δ = f (θc) (Fig.1, curve 1). It is possible to approximate this dependence with the function

    δ=3.15(1exp(θ/0.55)), (5)

    where δ=ER2BTs2Rgaexp(ETsRg) and θ=E(TCTS)Ts2Rg, Tc is the temperature at the center of the reactor in the steady-state mode, TS is the temperature of the reactor walls. Figure 1 also shows the relation between dimensionless temperature (θc) and δ for cylindrical reactors with h/d = 2 and h/d = 5 (Fig.1, curves 2 and 3, respectively). The parameters of the equation δ = f(θc) have been found for δδcr, i.e., for the case where critical conditions have not been reached. The critical values of the F-K parameter for cylindrical reactors with finite length can be obtained by solving the equation (4) numerically. In the extreme case of h d, the parameter δcr should approach the critical value for an infinite cylinder, δcr = 2 [19].

    Table 1. The values of the dimensionless temperature at the center of the reactor (θc) at different values of δ for a cylindrical reactor (h/d = 1).
    δ θC δ θC
    0 0 2.1 0.664
    0.25 0.052 2.2 0.7242
    0.5 0.1082 2.3 0.792
    0.75 0.1693 2.4 0.8704
    1 0.2365 2.5 0.9644
    1.25 0.3112 2.6 1.0842
    1.5 0.3956 2.65 1.1617
    1.75 0.4932 2.7 1.2615
    2 0.6098 2.75 1.4143
     | Show Table
    DownLoad: CSV
    Figure 1. The dependences for reactors with h/d ratios of (1) 1, (2) 2, and (3) 5.

    The explicit form of the equation F(δ, θc) = 0 in the unknown parameters E and B is obtained by substituting the expressions for δ and θ into Eq. (5). The temperature at the center of the reactor (Tc) when the steady-state mode is established should be estimated from the results of a thermal experiment under the conditions of a constant temperature at the walls (Ts). The experiment should be performed several times at different Ts values. In this way, the pairs of temperatures (Ts1, Tc1), (Ts2, Tc2), etc. are determined. Each (Ts, Tc) pair should be substituted into the equation F(δ, θc) = 0. This yields a set of simultaneous algebraic equations whose solution allows the apparent kinetic constants to be determined.


    3. Results and Discussion


    3.1. An Example of Calculation of the Kinetic Parameters

    Let us consider an example where the procedure described above is used to calculate the kinetic parameters E and B for a cylindrical reactor with h/d = 1. The values of the kinetic parameters can be determined from the results of two thermal experiments. The temperatures at the center of the reactor (Tc1 and Tc2) in the steady-state mode can be estimated by numerically solving Eq. (1). The calculations have been performed for a cylindrical reactor with equal height and diameter of h = d = 0.1 m. In this study, the thermophysical parameters have been taken from [17]: E = 101.7 kJ/mol and B = 1.1 * 1011 K/s, a = 1.5 •10-7 m2/s. In our calculations, the temperature at the sample surface has been taken to be Ts1 = 373.15K (100 °C) in the first case and Ts2 =383.15K (110 °C) (in the second case); in the center, the symmetry condition has been assumed to be satisfied. The calculated temperatures at the center of the reactor are Tc1 = 376.05K (102.9 °C) and Tc2 = 392.85K (119.7 °C), respectively.

    In order to determine the parameters E and B, let us rewrite Eq. (5) explicitly:

    ER2BTs2Rgaexp(ETsRg)=3.15(1exp(E(TCTS)0.55Ts2Rg)), (6)

    which allows us to substitute the temperatures (Ts1, Tc1) and (Ts2, Tc2). The result is a system of two equations with two unknowns, E and B. By dividing the first equation of the set of equations by the second we will get the equation in only one unknown E. It is convenient to solve this equation using a graphical calculation. Then we will evaluate the parameter B. The following kinetic parameters were obtained using this procedure for the aforementioned system of equations: E = 99.7 kJ/mol and B = 1 * 1011 K/s. The value of the activation energy obtained by solving the inverse problem differs from that used in calculating the temperature by about 1%.

    For a discriminating comparison between the temperatures used in Chen's method and in a proposed method we consider the temperature profiles in the reactor for the cases when the crossing point temperature occurs (Fig.2, curve 1) and when the steady state is established (Fig.2, curve 2) at 110 °C (Ts). As seen from Fig.2, in an experiment where critical conditions are not reached, the crossing point temperature only slightly differs from the temperature at the reactor wall, which is inconvenient for analysis of the experimental data. The temperature at the center of the reactor in the steady-state mode is much higher; therefore, it is more suitable to use this value as a characteristic temperature.

    Figure 2. Temperature profiles in thereactors when (1) the crossing point temperature occurs and (2) the steady state is established (the temperature at the surface was 110 °C).

    In our modified method for estimating the kinetic constants of coal oxidation, the temperature in the steady-state mode, where the critical conditions are not reached, serves as a characteristic temperature. One should only experimentally measure the temperature at the center of the sample and know the temperature conductivity coefficient to determine the unknown parameters.


    3.2. The Effect of Oxygen Consumption in Estimating the Kinetic Constants

    The solution of the heat conductivity Eq. (1) yields the relationships from which the kinetic constants can only be determined if the concentrations of the reagents are assumed to be constant. This limits the use of thermal methods based on the F-K analysis. To use this approximation, one should estimate the conditions of its applicability. For this purpose, let us recalculate the temperature in the sample during the heating, now with the oxygen consumption in the oxidation reaction taken into account. It is assumed that there is no forced convection, oxygen transport being accounted for by molecular diffusion.

    Let us consider heating of coal in a cylindrical basket-type reactor, which ensures air supply to the coal. Numerical calculations will be performed for a system with the same parameters as in the system considered above, where the oxygen expenditure was not taken into consideration. It is assumed that, at the initial moment of time, oxygen is evenly distributed in the pores between granules throughout the rector. The coal in the reactor is considered as a quasi-homogeneous medium with an effective coefficient of diffusion. The initial oxygen concentration in the sample (C0) is determined from the condition of equilibrium with the ambient air; the porosity factor is assumed to be 30%.

    Within the quasi-homogeneous approximation, the process of heating is described by the equations of thermal conduction and diffusion containing a reaction term; the reaction is assumed to be of the first order with respect to oxygen:

    ρCpTt=k2T+QAexp(ERgT)CC0,εCt=D2CAexp(ERgT)CC0. (7)

    Where C is the oxygen concentration (mol L−1) (it is taken into consideration that the pre-exponential factor includes the concentration C0); D is the diffusion coefficient in the porous medium, ε - porosity. The system of equations (7) was numerically solved under the boundary conditions of constant temperature Ts and oxygen concentration C0. In the calculations, the kinetic parameters of the coal from the study [17] (see above) was used; it is suggested that they correspond to the characteristic of coal saturated with oxygen. The temperatures at the surface of the sample were specified as Ts1 = 373.15 K (100 °C) and Ts2 = 383.15K (110 °C).

    The calculations have shown that, when the steady-state mode has been reached, the oxygen concentration in the center of the sample is decreased by about 4% in the first case and about 10% in the second case (Fig.3, curve 1). This oxygen "deficit" leads to a decrease in temperature. In the steady-state mode, the temperature is settled at Tc1 = 375.95K (102.8 °C) and Tc2 = 391.75 K (118.6 °C) in the first and second cases, respectively (Fig.3, curve 2). For comparison, the respective temperatures obtained for this system without oxygen consumption taken into account are 376.05K and 392.85K (see above).

    Figure 3. Changes in (1) the oxygen concentration and (2) the temperature at the center of the reactor with time (the temperature at the surface was 110 °C).

    Let us estimate how the decrease in the Tc-Ts value due to the decrease in oxygen concentration affects the values of the kinetic constants calculated by the modified method. For this purpose, let us use the temperatures (Ts1, Tc1) and (Ts2, Tc2) calculated with the oxygen consumption in the sample taken into account. The kinetic parameters determined by our method are E = 100 kJ/mol and B = 1.3·1011 K/s. This example shows that the consideration of the change in Tc due to oxygen consumption under the given conditions only slightly affects the values of the kinetic constants calculated by the modified method. In the general case, a thermal experiment should be carried out under the conditions that ensure a negligibly small consumption of the reagents.


    4. Conclusion

    The modified method proposed here allows the kinetic constants of coal oxidation to be determined from the temperatures at the center and surface of the sample in a steady-state mode, where the critical conditions are not reached. The method is based on the dependence of the dimensionless temperature on the F-K parameter. This approach makes it possible to substantially reduce the number of experiments compared to the traditional steady-state method. The kinetic constants evaluation procedure was shown for the case of numerical experiment (as an example).


    Acknowledgments

    The work was supported by the Federal Target Program "Research and development on priority directions of Russia scientific-technological complex for 2014-2020", event 1.2. Unique identifier of the project RFMEF157514X0062.


    Conflict of Interest

    All authors declare no conflicts of interest in this paper


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