HYBRID SIMULATION OF THE SLOW DISCHARGE PROCESS FROM ADSORBED METHANE TANKS

Transcription

1 Eurotherm Seminar N 81 Reactive Heat Transfer in Porous Media, Ecole des Mines d Albi, France June 4 6, 2007 HYBRID SIMULATION OF THE SLOW DISCHARGE PROCESS FROM ADSORBED METHANE TANKS Silvia da Costa Hirata Mechanical Eng. Dept., Universidade Federal do Rio de Janeiro, COPPE/UFRJ, Brasil silvia@fast.u-psud.fr Paulo Couto Industrial Eng. Dept., Universidade Federal do Rio de Janeiro, POLI/UFRJ, Brasil pcouto@petroleo.ufrj.br Luciano G. Lara Petrobras Petróleo Brasileiro S.A., IEABAST/EAB/ENPRO, Rio de Janeiro, Brasil lglara@petrobras.com.br Renato Machado Cotta Mechanical Eng. Dept., Universidade Federal do Rio de Janeiro, POLI/COPPE/UFRJ, Brasil cotta@mecanica.coppe.ufrj.br ET Abstract. The Integral Transform Method is employed in the hybrid numerical-analytical solution of the energy equation for the slow discharge of natural gas (modeled as pure methane) tank filled with porous carbonaceous adsorptive material. A transient one-dimensional nonlinear formulation is adopted, which includes the compressed and adsorbed natural gas thermal capacitance, the reservoir wall thermal capacitance effect and the gas compressibility influence. The overall mass balance is employed to determine the pressure field evolution, here assumed as spatially uniform. A thorough covalidation analysis is performed, with both the Mathematica system NDSolve numerical solution for a special case, and a previously proposed finite differences implementation for the two-dimensional case. The relative importance of some terms in the energy equation formulation, as presented in different contributions in the literature, is then critically inspected. Keywords. Natural gas, adsorption, ANG, hybrid methods, integral transforms. 1. Introduction Natural gas is becoming an economically attractive fuel for many industrial applications around the world. However, the use of natural gas is still limited since the storage and transportation of natural gas are not convenient when compared to conventional liquid fuels. When natural gas is in the gas phase, transportation occurs in gas pipelines and storage is performed in high-pressure cylinders, as CNG (compressed natural gas) at pressure levels around 200 atmospheres. Although this technique provides storage capacity of 200 V/V (volume of stored natural gas, measured at Standard Pressure and Temperature conditions STP, per unit storage volume), it requires reliable and safe cylinders as well as costly compressing equipment. If the natural gas is in the liquid phase, as liquefied natural gas LNG, volumes as large as 600 V/V can be transported in ships and trucks. In this case, the gas must be cooled to a temperature as low as 160ºC. This technique is highly energetically expensive as heat leaks during transportation and storage vaporizes the LNG. Also, to be reused, the gas must be vaporized in appropriate equipment. Adsorbed natural gas ANG has recently emerged as an alternative to CNG and LNG. Adsorption is the gas molecules uptake process in an interfacial layer (usually a highly porous media), either by capillary condensation or by Van der Walls forces. In the case of natural gas, the uptake process occurs at relatively low pressures, in the order of 35 to 50 atmospheres, which provides three main benefits: (1) compression costs reduction, (2) increased safety of the reservoir, and (3) design flexibility of the storage tanks. Despite of these benefits, the storage capacity of ANG reservoirs is lower than LNG and CNG, being of the order of 90 V/V (Brady et al., 1996) up to 164 V/V (Inomata et al., 2002). The U.S. Department of Energy storage target for ANG vessels was set at 150 V/V and it was later revised to 180 V/V. According to Biloé et al. (2001), the performance and viability of an ANG system depends closely on the microporous characteristics of the adsorbent as well as on the heat and mass transfer properties. The adsorption phenomenon that occurs during the charge of an ANG reservoir is an exothermic process. The increase in temperature, caused by the release of latent heat of adsorption, results in less stored methane capacity under dynamic conditions. During discharge of the ANG reservoir, the desorption process is endothermic, which causes the adsorbent bed to cool down. This causes some amount of gas to be retained inside the reservoir at depletion pressure, reducing the delivered methane capacity. The thermal effects depend closely on the heat transfer properties of the adsorbent bed as well as on the external heat exchange at the vessel wall (Chang & Talu, 1996; Mota et al., 1997 and Vasiliev et al., 2000).

2 The natural gas admission in a power cycle is controlled by power requirements of the system. In this case, discharge time is increased considerably by the slow rate of natural gas discharge, which usually occurs at a constant flow rate outward the cylinder at the opening (Mota et al., 1997 and Vasiliev et al., 2000). The heat of adsorption during this process decreases the temperature of the adsorbent bed, which in turn increases the amount of natural gas retained inside the cylinder at depletion (Biloé et al., 2001). However, the discharge process is not adiabatic, and the heat transferred from the surroundings to the cylinder partially compensates this cooling effect. Although the literature presents suggestions to improve the storage capacity of ANG systems, the thermal control of an adsorptive reservoir was barely studied. The addition of thermal energy to the adsorbent bed increases its temperature, enhancing the natural gas deliverance capacity during the discharge process. Based on a model previously established for the analysis of dynamic thermal processes in ANG tanks (Mota et al., 1997), a finite differences procedure was implemented for the two-dimensional analysis of the slow discharge process (Lara, 2005, Lara et al., 2006a, Couto et al., 2006) and employed in the analysis of a heat pipe solution for its thermal control (Lara et al., 2006b). The present work will focus on the development of an error controlled solution of the energy equation related to the slow discharge process of adsorbed methane in cylindrical tanks filled with porous adsorptive material. The aim is to provide a benchmark solution for validation of all such previous developments, while offering an alternative solution path for the inspection of the different modeling and formulations so far considered in the open literature, which are indeed not unanimous on which terms to account for. For this purpose, the Generalized Integral Transform Technique (GIIT) (Cotta, 1990, Cotta, 1993, Cotta & Mikhailov, 1997, Cotta & Mikhailov, 2006) is employed in the hybrid numerical-analytical solution of the proposed one-dimensional transient formulation for the energy equation that governs the gas temperature inside the tank, including a fifth kind boundary condition that accounts for the wall thermal capacitance effect. An implicit filtering procedure is devised to completely eliminate the transient operator from such boundary condition, relaxing the need for employing other more cumbersome convergence acceleration techniques. Also, in order to avoid a coupled coefficients matrix in the transient operator of the transformed energy equation, all the nonlinearities of the problem are moved into the general source term, which also includes the heat of adsorption and compressibility effects. The solution procedure was entirely developed in the Mathematica v. 5.2 system (Wolfram, 1999), including all of the symbolic computation steps required by the hybrid solution implemented. Numerical results were first critically covalidated with a numerical solution obtained by the Method of Lines as implemented in the routine NDSolve (Wolfram, 1999), for a simplified situation without the wall capacitance effect and for a prescribed pressure field evolution. In addition, the hybrid solution is compared with the previously developed two-dimensional finite differences solution (Lara, 2005, Couto et al., 2006) and other numerical implementations available in the literature (Mota et al., 1997). 2. Problem Formulation A nonlinear one-dimensional transient model for the discharge process of a cylindrical ANG vessel is presented. A sufficiently long cylindrical vessel of external radius R ext and length L, with a wall thickness of δ w is considered. The gas flows outwards from the cylinder through a cylindrical opening at the origin of the system, at a constant mass flow rate. Heat transfer by natural convection (h ) between the external walls of the cylinder and the surroundings at T is considered. The activated carbon bed is considered to be composed of activated carbon pellets with an effective porosity of j eff. A temperature axisymmetric condition is considered at the centre of the cylinder. The energy balance equation derived for the slow discharge process was obtained by writing the first law of thermodynamics for transient regime (Lara, 2005), accounting for the contribution of the solid phase (adsorbent), gas phase (adsorbate), and the adsorbed phase in the total energy stored in the reservoir. Also, the contribution of the energy transfer by heat diffusion in the solid phase is considered. The latent heat of adsorption is considered as a rate of internal heat generation. The assumptions made for the energy conservation equation derivation are: (1) instantaneous phase equilibrium between the adsorbed phase and the gas phase within the porous structure, (2) the local temperature of the adsorbent and free gas volume is the same due to a high intensity heat transfer between solid and gas phases. Also, (3) for the solid and adsorbed phases, the internal energy depends only on the local thermodynamic temperature and (4) the adsorbed phase is considered incompressible. As the discharge process is considered to be slow, (5) the pressure gradient inside the cylinder is negligible (Mota et al., 1997), i.e, the pressure is a function of time, only. According to these assumptions, the energy equation is written as: Trt (,) 1 T dpt () q ( ϕeff cp, g ρg + ρscp, s + ρscp, g q) = keff r + ϕeff ρsδh, t r dt t in 0 < r <R int, t > 0 (1) The left hand side of Eq. (1) accounts for the energy stored in the control volume in the compressed gas within the pores of the porous structure, activated carbon bed and adsorbed phase, respectively. The first term in the right hand side of Eq. (1) represents the heat diffusion in the activated carbon bed, while the last two terms represent the compressibility effect of the gas and the heat of adsorption. These two terms act as a heat source term in the energy equation. Initially, the system is considered to be at a homogeneous and uniform temperature and pressure, given by T 0 and P 0 respectively. The initial and boundary conditions of Eq. (1) are:

3 Tr (,0) = T0, in 0 < r <R int, (2.a) T(0, t) = 0, t > 0 T( Rint, t) T keff + ρwcp, wδw = h ( T T), at r = R, t int 0 (2.b) t > (2.c) Boundary conditions (2.c) account for the thermal capacity of the cylinder wall (ρ w c p,w ) and for the heat transfer by natural convection at the external wall of the cylinder. The heat transfer coefficient h can be obtained from natural convection correlations for horizontal cylinders available in the literature. The mass conservation equation is given by: 0 R ( eff g + s ) int 2πL ϕ ρ ρ q rdr = m t (3) The first term inside the integrals represents the mass of compressed gas phase in the pores of the adsorbent bed, while the second term represents the mass of the adsorbed gas phase. Equation (3) provides an expression for the total variation of the gas mass inside the reservoir as a function of time and of the constant mass flow rate. By using the ideal gas equation to correlate P, T and ρ g, Eq. (3) can be solved to determine the pressure history as a function of time [P = f(t)] inside the cylinder for a given mass flow rate. The amount of gas adsorbed by an adsorbent material when equilibrium is reached for a given temperature and pressure, is a function of the nature of the gas to be adsorbed and of the adsorbent material. For a given adsorbateadsorbent system, the adsorption equilibrium relation is expressed by: q = f( P) T (4) where q is the amount of gas adsorbed per unit mass of adsorbent (commonly in kg gas /kg solid ). For the adsorption of natural gas, the gas is considered uniform and homogeneous, with thermodynamic properties well defined by its temperature and pressure. Mota et al. (1997) present the following experimentally obtained Langmuir equilibrium relation for the adsorption of methane in a G216 activated carbon bed: q = ( q bp)/(1+ bp) (5) m where q m = 55920T 2.3 and b = exp(806/t). 3. Solution Methodology The Generalized Integral Transform Technique (GITT) is a hybrid numerical-analytical approach for nonlinear diffusion and convection-diffusion problems, that was originated form the classical integral transform method for linear transformable diffusion problems (Cotta, 1993, Cotta & Mikhailov, 1997). The basic concept is to propose eigenfunction expansions in all but one independent variable, which results in an analytical representation of the desired potential in all but this only one variable. Upon integral transformation of the original partial differential system, a coupled system of ordinary differential equations for the transformed potentials then results, which is numerically solved to yield the expansion coefficients. The main features of this approach is the automatic error control procedure, the direct application to nonlinear formulations and irregular geometries, and the mild increase in computational cost for multidimensional applications. Its application to the present class of problems is particular motivating in light of the strong nonlinear nature of practically all of the terms and the relative importance of the source terms. The first step in the application of the integral transform method in the present problem is to rewrite Eq. (1) so as to incorporate all of the nonlinear terms in the right hand side of the energy equation, thus leaving the transient operator available for exact integral transformation, which avoids an implicit transformed system that would require matrix inversion for each step of the numerical integration of the transformed ODE system. Since the adsorbed phase concentration, q, can be rewritten as a function of temperature and pressure, its derivative can be rewritten as: q( T, P) q( T, P) T( r, t) q( T, P) dp( t) = + t T t P dt Then, Eq. (1) can be rewritten as: qt (, P) T( rt, ) 1 T qt (, P) dpt ( ) ( ϕeff cp, g ρg ρscp, s ρscp, g q ρs H ) k eff r ϕeff ρs H Δ = + Δ, T t r P dt in 0 < r < R, t > 0 int (6) (7)

4 where the required derivatives of q(t, P) are symbolically obtained from Eq. (5). After labeling the nonlinear term in the transient term as an effective thermal capacitance, ρc p,m (T, P), the energy equation is given by: Trt (,) 1 T qtp (, ) dpt () = keff r + ϕeff ρsδ H / ρcp, m ( T, P), in 0 < r < Ri, t t r P dt nt > 0 (8) The next step is the proposition of a filtering solution that explicitly accounts for the transient operator in the boundary condition (2.c), that refers to the wall thermal capacitance effect. The temperature field is then written as: Trt (,) Tf (,) rt T(,) rt = + (9) For the purpose of constructing a filter T f, we consider a simple steady heat conduction formulation that obeys the original boundary conditions, in the form: T f r = 0, r in 0 < r < R int (10.a) Tf (0, t) = 0, t > 0 (10.b) T f ( int, ) keff + ρwcp, wδ w = h ( T Tf ), t at r = Rint, t > 0 (10.c) This implicit filter is readily obtained as: ρwcp, wδw T( Rint, t) Tf () t = T h t The proposed filter is r-independent and eliminates the transient operator in the boundary condition of the partial differential system, but it is not known a priori, since it depends on the unknown potential at the wall, T (R int, t). Substitution of this filter into the original system, Eqs. (8, 2.a-c), results in the filtered problem below: T (,) r t 1 1 T q( T, P) dp() t dtf () t = keff r + ϕeff ρsδh, in 0 < r < Rint, t > 0 (12.a) t ρcpm, ( T, P) r P dt dt T r T0 T f (,0) = (0), in 0 < r <R int, (12.b) T (0, t) = 0, t > 0 T ( Rint, t) keff + h T ( Rint, t) = 0, (11) (12.c) t > 0 (12.d) Equation (11) can be interpreted itself as the differential equation for T f (t) that couples with the above partial differential formulation, in the form: dt f () t T ( Rint, t) h + = ( T Tf () t dt t ρwcp, wδw with the initial condition ) (13.a) ρ c δ 1 qt (, P) dp(0) T T H h c T P P dt w p, w w 0 0 f (0) = ϕeff ρsδ ρ pm, ( 0, 0) (13.b) The proposed eigenfunction expansion is based on the following eigenvalue problem: 1 ψ i () r 2 r + μψ i i() r = 0, in 0 < r < int r R (14.a) ψ i () r = 0, r = 0 (14.b)

5 ψ i ( Rint ) keff + h ψ i ( Rint ) = 0, t >0 (14.c) where ψ i s are the eigenfunctions, μ i s are the eigenvalues, and the normalization integral is written as: int 2 N = rψ () r dr (14.d) i 0 R i while the normalized eigenfunctions and eigenvalues are obtained from ψ i() r J0( μ r) ψ i () r = = i (14.e) N N i keff μj1( μrint ) + h J0( μrint ) 0 i The integral transform pair can then be defined as: = (14.f) Rint Ti() t = rψ i( r) T ( r,) t dr, 0 transform (15.a) T (,) r t = ψ i() r Ti(), t inversion (15.b) i= 1 Rint The partial differential equation (12.a) is now integral transformed, by operating it with rψ i () r dr, to yield: 0 dt () () i t dtf t = hi( t) fi, for i = 1,2,..., dt dt t > 0 (16.a) R int 1 1 T q( T, P) dp( t) hi() t = rψ i( r) keff r + ϕeff ρsδh dr 0 ρcpm, ( T, P) r P dt (16.b) Rint fi = rψ i() r dr 0 (16.c) ( 0 f ) Ti(0) = T T (0) f i (16.d) The coefficients vector h i (t) needs to be evaluated numerically due to the nonlinear nature of this term. Here, however, a semi-analytical procedure was preferred (Cotta & Mikhailov, 2005), based on the exact integration of the periodic behavior of the eigenfunctions. For a given pressure evolution, system (16) is then joined with Eqs. (13) and corresponding initial conditions, to form the ODE transformed system which is numerically integrated employing the NDSolve function of the Mathematica system (Wolfram, 1999). The inversion formula (15.b) followed by the filtering expression, Eq. (9), then recover the desired temperature field in analytical form along the radial coordinate. An iterative procedure is finally required for updating the pressure field evolution, P (t). The continuity equation can be integrated by the same semi-analytical approach employed for the transformed system coefficients (Cotta & Mikhailov, 2005), yielding a differential equation for the pressure field in terms of the transformed potentials above described. The initial estimate for the pressure evolution is then provided by the isothermal discharge limiting solution. 4. Results and Discussion Before presenting and discussing some physical aspects of the problem, we first seek a verification of the constructed symbolic-numerical code and a demonstration of the performance of the proposed methodology. Mota et al. (1997), presented a one-dimensional transient theoretical analysis of the slow discharge of a 50 litres ANG cylinder, where emphasis was given to thermal and mass diffusion effects in the activated carbon bed. Natural gas was modelled as ideal gas (pure methane). Although Mota et al. (1997) presented a complete energy equation, the pressure history was imposed, and not determined by the mass conservation equation. The imposed pressure history was expected to lead to a constant mass flow rate at the cylinder outlet, but it is not explicitly provided which makes it more difficult to reproduce the results in their work. Nevertheless, due to the similarities between the two models, we have followed the work of (Mota et al., 1997) in this comparison effort, and the following pertinent data was employed:

6 Table 1. Parameters used for code validation and solution methodology demonstration Parameter Mota et al. (1997) Activated carbon bed properties: Type G216 carbon pellets Density ρ s 410 kg/m 3 Specific heat c p,s 650 J/kg.K Effective conductivity k eff 1.2 W/m.K Effective porosity ϕ eff 0.74 Cylinder geometry: Material Stainless steel Density ρ w 7900 kg/m 3 Specific heat c p,w J/kg.K Internal radius R int 0.14 m Length L 0.85 m Thickness δ w 0.01 m Operational conditions: Initial pressure P MPa Depletion pressure P d MPa Initial temperature T K Ambient temperature T Heat transfer coefficient h K W/m 2 C Flow rate m kg/min (1) Adsorption characteristics: Heat of adsorption ΔH J/kg 1 Value inferred from performance coefficient data provided by Mota et al. (1997). Therefore, for an estimated pressure history obtained from a sixth order polynomial fit, presented in Figure 1 below, we have first compared the GITT solution with the NDSolve routine, available in the Mathematica system (Wolfram, 1999) for the case of negligible wall thickness, with excellent agreement. For the more general case of a participating wall, the built in routine of Mathematica based on the Method of Lines cannot possibly handle the partial differential equation with the time operator in the boundary condition. In this case, we have employed the GITT solution for the wall temperature time variation, to feedback the NDSolve routine and obtain some approximate estimates for comparison purposes. One such set of results is presented in Figure 2, where the time history for the center (blue lines) and wall (red lines) temperatures from the two approaches are compared, with the GITT solution presented in dashed lines and the NDSolve approximation in solid lines. Next, we conduct a critical comparison of the GITT solutions without and with wall conjugation effects, by reducing the wall thickness to a negligible value. Figure 3 shows various radial temperature profiles, for different times along the discharge process (t = 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0 hours). Earlier times are represented by predominantly red colors and later times with predominance of the blue color. The dashed lines are for the results with wall participation, and the solid lines are for the case accounting for the wall thermal capacitance. Clearly, the wall capacitance has a marked effect in slowing down the cooling down of the porous bed along the slow discharge process, especially in regions closer to the wall. Nevertheless, the center wall temperature is more than 10 C higher after 2 hours of discharge when considering the wall effect, also suggesting the adoption of slender cylindrical tanks. Figure 1 Estimated discharge process pressure history for comparison with (Mota et al., 1997) Figure 2 Comparison of center (blue) and wall (red) temperature evolutions with NDSolve (solid) and GITT (dashed) solutions.

7 Figure 4 then shows a comparison of these same GITT results with wall participation against the numerical results of (Mota et al., 1997), shown as dots, where one may observe a quite reasonable agreement for shorter times and an increasing discrepancy for larger discharge times. Figure 3 Comparison of radial temperature profiles (GITT) at different times during discharge (from red to blue), without (dashed) and with (solid) wall thermal capacitance effects. Figure 4 Comparison of radial temperature profiles (GITT) at different times along the discharge (from red to blue), with the results (dotted) of Mota et al. (1997). The present research effort has already resulted in the consolidation of a two-dimensional model for the ANG discharge process that includes the pressure evolution determination for an imposed mass flow rate and the adoption of a thermal control device at the center of the tank, such as a heat pipe (Couto et al., 2006, Lara et al., 2006a, Lara et al. 2006b). It is therefore of interest to covalidate these two implementations so as to provide a more definitive confidence on such simulation efforts for future comparisons with experimental results and design of charge and discharge cycles in actual applications. Figure 5 below presents the GITT results for the radial temperature profiles, for both the case of an imposed pressure gradient (solid), such as in Fig.1, and for a constant mass flow rate (dashed), as chosen in Table 1, when the pressure history is automatically determined by the algorithm. In addition, the results from (Couto et al., 2006) for a fixed mass flow rate are presented as dots, demonstrating the excellent agreement with the equivalent results by the Generalized Integral Transform Technique. It can also be noticed that the estimated pressure history from the plotted results of (Mota et al., 1997) still leads to reasonably accurate results for the temperature profiles, as illustrated by the dashed and solid lines, and that this aspect could not explain the increasing discrepancies between the present results and those of Mota et al. (1997) for the larger discharge times. Finally, in order to analyse the influence of the wall heat transfer coefficient, we increase its value by 2 orders of magnitude. The variation of h greatly inlfuences the temperature profiles, as shown in Figure 6, where the dashed lines represent the profiles obtained with h = 400 W/m 2 K (forced convection), while the continuous lines were obtained with h = 4 W/m 2 K (natural convection). As we increase h, the convection heat transfer is enhanced at the cylinder surface, inducing an augmentation of 9 C at the center and of 19 C at the cylinder wall. Actually, for h = 400 W/m 2 K, the wall temperature remains approximately with the same initial value throughout the discharge process. Figure 5 Comparison of radial temperature profiles (GITT) at different times during discharge (from red to blue), without (dashed) and with (solid) prescribed pressure history, and the results (dotted) of Couto et al., Figure 6 Comparison of radial temperature profiles (GITT) at different times along the discharge (from green to blue), with h = 400 W/m 2 K (dashed) and h = 4 W/m 2 K (solid).

8 5. Conclusions This work presented a hybrid numerical-analytical solution based on the Generalized Integral Transform Technique (GITT) of a one-dimensional transient non-linear model for the slow discharge process of adsorbed natural gas cylinders. Natural gas was modelled as pure methane with an ideal gas thermodynamic behaviour. The model accounts for the thermal capacity of the cylinder wall, convective heat transfer from the cylinder wall and the surrounding environment, and a prescribed mass flow rate leaving the cylinder. The data obtained with the presented hybrid solution was compared to numerical data available in the literature (Mota et al., 1997) and the results of this comparison were encouraging, with the radial temperature profile presenting the same trends. The main difference between the present model and the one described by Mota et al. (1997) is that the later considered a known pressure history, while the former may also obtain the pressure history from the mass balance equation. This seems to be a more realistic condition as the consumption of natural gas from a cylinder is dictated by the power requirements of a thermal engine or cycle. The model was then used to co-validate the finite differences solution of a two-dimensional transient model previously developed (Lara, 2005; Lara et al., 2006a and 2006b, and Couto et al., 2006), with excellent agreement. The proposed solution was also employed in the critical inspection of the wall thermal capacitance effect and of the wall heat transfer coefficient with the external environment. 6. Acknowledgements The authors wish to acknowledge the financial support provided by CNPq/CTPETRO, through projects no /04-7 and no / References Biloé, S.; Goetz, V.; and Mauran, S., Dynamic Discharge and Performance of a New Adsorbent for Natural Gas Storage. In: AIChE Journal, Vol. 47, no. 12, pp Brady, T. A.; Rostam-Abadi, M.; and Rood, M. J., Applications for Activated Carbons from Waste Tires: Natural Gas Storage and Air Pollution Control. In: Gas Separation and Purification, Vol. 10, no. 10, pp Chang, K. J. and Talu, O., Behaviour and Performance of Adsorptive Natural Gas Storage Cylinders During Discharge. In: Applied Thermal Engineering, Vol. 16, no. 5, pp Cotta, R.M., 1990, "Hybrid Numerical-Analytical Approach to Nonlinear Diffusion Problems", Num. Heat Transfer, Part B, V. 127, pp , Cotta, R.M., 1993, Integral Transforms in Computational Heat and Fluid Flow, CRC Press, Boca Raton, FL. Cotta, R.M., and Mikhailov, M.D., 1997, Heat Conduction: Lumped Analysis, Integral Transforms, Symbolic Computation, Wiley-Interscience, New York. Cotta, R.M., and M.D. Mikhailov, 2005, Semi-Analytical Evaluation of Integrals for the Generalized Integral Transform Technique, Proc. of the 4 th Workshop on Integral Transforms and Benchmark Problems IV WIT, CNEN, Rio de Janeiro, RJ, August. Cotta, R.M., and M.D. Mikhailov, 2006, Hybrid Methods and Symbolic Computations, in: Handbook of Numerical Heat Transfer, 2 nd edition, Chapter 16, Eds. W.J. Minkowycz, E.M. Sparrow, and J.Y. Murthy, John Wiley, New York, pp Couto, P., L.G. Lara, D.M.A. Sophya, and R.M. Cotta, Thermal Control of Adsorbed Natural Gas Reservoirs under Discharge Dynamic Condition, Proc. of the 13th Int. Heat Transfer Conference, Sidney, Australia, Inomata, K.; Kanazawa, K.; Urabe, Y.; Hosono, H. and Araki, T., Natural gas storage in activated carbon pellets without a binder. In: Carbon, Vol. 40, pp Lara, L.G., Theoretical Analysis of the Discharge Process of Adsorbed Natural Gas Reservoirs (in Portuguese).M.Sc.thesis, Mech.Eng.Program, COPPE, Fed.Univ.of Rio de Janeiro, Brazil. Lara, L.G., P. Couto, R.M. Cotta, and D.M.A. Sophia, Theoretical Model for the Discharge Process of Adsorbed Natural Gas Reservoirs, 11th Brazilian Congress of Thermal Sciences and Engineering, ENCIT 2006, Curitiba, Brazil, Paper no. CIT , December 2006a. Lara, L.G., P. Couto, R.M. Cotta, and D.M.A. Sophia, Gas Discharge Capacity Intensification of Adsorbed Natural Gas Reservoirs, 11th Brazilian Congress of Thermal Sciences and Engineering, ENCIT 2006, Curitiba, Brazil, Paper no. CIT , December 2006b. Mota, J. P. B.; Rodrigues, A. E.; Saatdjian, E.; and Tondeur, D., Dynamics of Natural Gas Adsorption Storage Systems Employing Activated Carbon. In: Carbon, Vol. 35, no. 9, pp Vasiliev, L. L.; Kaninchik, L. E.; Mishkinis, D. A.; and Rabetsky, M. I., Adsorbed Natural Gas Storage and Transportation Vessels. In: Int. J. of Thermal Sciences, no. 39, pp Wolfram, S., 1999, The Mathematica Book, 4th ed., Wolfram Media, Cambridge.