Improved lumped-capacitance model for heat and mass transfer in adsorbed gas discharge operations
Highlights
► An improved-lumped formulation for adsorbed gas storage is developed. ► The formulation can be used for a wider range than classical lumped models. ► A comparative analysis of different approximation schemes is presented.
Introduction
Adsorbed gas (AG) storage has been considered as a potential alternative to compressed gas (CG) or liquefied gas (LG) storage due to a number of factors, such as high pressures required for CG and low temperatures associated with LG. The AG storage option has been considered for a number of gases, such as methane [1], natural gas [2], [3], [4], hydrogen [5], [6], biogas [7], carbon monoxide and dioxide [8], [9], oxygen and nitrogen [8], [10]. In spite of the advantages associated with this storage mode, AG requires efficient thermal designs for their storage vessels, in order to properly minimize the limitations brought in by the heating and cooling effects inherent to the adsorption process. As a result, a number of mathematical models and computational simulations have been proposed as design tools for adsorbed gas reservoirs. The works of Barbosa Mota et al. [1], [11] provided a formulation for evaluating charge and discharge dynamics of adsorbed methane. Bastos-Neto et al. [12] considered a model similar to the one used in [11]; however, no intra-particle diffusion effects were considered. Another similar heat and mass transfer model was employed by Basumatary et al. [13]; however, in that study, the thermal conductivity of the adsorbent medium was allowed to vary with temperature. Walton and LeVan [14] presented a one-dimensional mathematical model for examining the influence of non-isothermal effects and impurities contained in natural gas during storage-cycles, and Zhou [15] employed a simple lumped formulation for natural gas storage. Hirata et al. [16] presented a one-dimensional model for gas discharge and employed a hybrid analytical-numerical methodology for the solution of the governing equations. A one-dimensional formulation was also employed by Santos et al. [17]. Recently, a lumped-capacitance formulation for adsorbed gas storage, developed in a fully dimensionless framework, was presented in [18].
A common agreement among previous studies is that typical discharge operations for fuel consumption applications involve lower gradients and can hence be simulated using one-dimensional, and in some cases, even lumped-capacitance formulations. Naturally, whenever possible, lumped formulations are preferred, due to their simplicity and consequently a lower required computational time. Very recently, the suitability of using a lumped-capacitance formulation for describing adsorbed gas discharge operations was analyzed [19], showing that, in some circumstances, the usage of lumped models can lead to a considerable error. These critical situations involve cases where the classical lumped approximation, in which small gradients are assumed, cannot be adopted. This problem may be circumvented by employing the so-called improved-lumped formulations, which remain simple, due to their independence on spatial coordinates, while retaining some information regarding gradients and boundary conditions through additional relations. One such methodology for obtaining improved-lumped formulations is based on the Coupled Integral Equations Approach (CIEA). The approach is based on the approximation of integrals in terms of a linear combination of the integrand values and its derivatives at the integration limits, an idea originally developed by Hermite [20] and first presented by Menning et al. [21]. The CIEA was successfully utilized in a variety of problems such as phase change problems [22], heat transfer in fins [23], heat exchangers [24], linear heat conduction [25], hyperbolic heat conduction [26], radiative cooling [27], ablation [28], drying [29], thermal analysis of nuclear fuel rods [30], heat conduction with temperature-dependent conductivity [31], and combined convection–radiation cooling [32]. In this context, the purpose of this study is to provide an improved-lumped formulation, using the CIEA, for simulating adsorbed gas storage operations. The text will focus on discharge operations and different levels of approximation will be presented and comparatively analyzed.
Section snippets
One-dimensional formulation
The problem of adsorbed gas discharge at constant mass flow rate, as usual for fuel consumption applications, can be described by a one-dimensional formulation within a cylindrical domain, as presented in Fig. 1. The governing equations for this problem, in dimensionless form are given by [33]:for , where and Ce is given by:
The dimensionless
Lumping procedure
The first step in the lumping process is defining the dimensionless volume average:where ϕ is an arbitrary quantity. Then, the energy equation is written in conservative form:
Operating equations (1a), (9) with the integral operator gives:
Coupled integral equations approach
The basis for the Coupled Integral Equations Approach (CIEA) is the Hermite approximation of an integral, denoted, , which is given by the general expression:where,and and its derivatives are defined for all . is the error in the approximation. This integration formula can provide a variety of approximation levels; however, since
Results and discussion
After the presentation of the CLSA and improved-lumped formulations for AG storage, simulation results are presented for illustrating the applicability of proposed formulations and approximation schemes. All results are calculated for a constant outlet mass flow rate, obtained for a situation with small pressure gradients, as previously considered for discharge operations for fuel consumption applications [11], [16]. This is achieved out by simulating cases with a high value. All results are
Conclusions
This paper proposed an improved-lumped formulation for heat and mass transfer in adsorbed gas storage operations. A case of slow discharge at a constant mass flow rate, which is common for fuel consumption applications and well documented in the literature, was analyzed. The improved-lumped model was based on averaging the transport equations of a one-dimensional formulation and employing the CIEA for obtaining relations between space-averaged quantities and boundary conditions. Two
Acknowledgments
The authors would like to acknowledge the financial support provided by the Brazilian funding agencies CNPq, CAPES, FAPERJ, as well as Universidade Federal Fluminense.
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